Advanced Search

Rouben Rostamian

8646 Reputation

23 Badges

19 years, 295 days
Contact Rouben Rostamian

MaplePrimes Activity


These are answers submitted by Rouben Rostamian

Maybe this?...

Rouben Rostamian 8646
November 30 2022
3 1

I am not familiar with Mathematica, so I am just guessing from your example what the Cases command is expected to do.  It appears that it picks all instances of sin(...) in a given expression.  Here is how that is done in Maple:

expr := [sin(x)/(sin(2-x)+1)-12, sin(x/2)^2, cos(x)];
indets(expr, 'specfunc(sin)');

That yields {sin(x), sin(-2 + x), sin(x/2)}.

Solution via finite differences...

Rouben Rostamian 8646
November 26 2022
0 2

This site is on the blink again and won't let me display my worksheet.   Here is the link to a worksheet that shows how to solve the system of PDEs through finite differences.

Download strange-PDE.mw

Use this model as needed...

Rouben Rostamian 8646
November 25 2022
1 0
> 

restart;
> 

with(plots):
> 

A1 := proc(t)
        pointplot([t,t], symbol=solidcircle, symbolsize=50, color=red);
end proc:
> 

A2 := proc(t)
        pointplot([t,t], symbol=solidcircle, symbolsize=50, color=blue);
end proc:
> 

display([
        animate(A1, [t], t=-1..0),
        animate(A2, [t], t=0..1)
], insequence);

 

 

Download mw.mw

 

A bug in pdsolve()...

Rouben Rostamian 8646
November 25 2022
4 6

Your equations are uncoupled.  You may call pdsolve() to solve each of them separately.  That works on all versions of Maple that I have tried.

In principle, pdsolve() should be able to solve the two (uncoupled) equations together as a system, but that fails in Maple 2022 as we see in the attached worksheet.

Here is how things work on solving the system in various versions of Maple:

  • Maple 2017 and 2018 return no solutions, and no errors;

  • Maple 2019, 2020, 2021 return the correct solution;

  • Maple 2022 fails on error.

> 

restart;
> 

kernelopts(version);

`Maple 2022.2, X86 64 LINUX, Oct 23 2022, Build ID 1657361`

> 

Physics:-Version();

`The "Physics Updates" version in the MapleCloud is 1354. The version installed in this computer is 1342 created 2022, November 2, 11:29 hours Pacific Time, found in the directory /home/rouben/maple/toolbox/2022/Physics Updates/lib/`

Pdsolve gets the correct solution here:

> 

pde1 := diff(u(x,t),t) = 0;
ic1 := u(x,0) = f(x);

diff(u(x, t), t) = 0

u(x, 0) = f(x)

> 

pdsolve({pde1,ic1});

u(x, t) = f(x)

Pdsolve gets also the correct solution here:

> 

pde2 := diff(v(x,t),t) = 0;
ic2 := v(x,0) = g(x);

diff(v(x, t), t) = 0

v(x, 0) = g(x)

> 

pdsolve({pde2,ic2});

v(x, t) = g(x)

BUG: Pdsolve gets confused when solving the two (uncoupled) equations together:

> 

pdsolve({pde1, pde2, ic1, ic2});

Error, (in pdsolve) invalid input: indets expects 1 or 2 arguments, but received 3

> 

 

Download mw.mw

Note added Nov 30, 2022

The outcome of the discussion that followed this post was that Maple 2022's pdsolve() was not at fault after all.  The trouble was due to a not-up-to-date version of the Physics package on my machine, and it was resolved after upgrading to the latest version of the Physics package.

 

Getting started...

Rouben Rostamian 8646
November 18 2022
0 0

I am not going to do the entire homework for you but here is something to get you started.
 

> 

plot(8+8*sin(theta), theta=0..2*Pi, coords=polar, scaling=constrained);

 
> 

plot3d(25 - x^2, x=0..5, y=1..8, view=[0..5,0..8,default]);

 
> 

plots:-display(
    plot3d([10*cos(t)*cos(s), 10*cos(t)*sin(s), 10*sin(t)], t=-arccos(6/10)..arccos(6/10), s=-Pi..Pi),
    plot3d([6*cos(t), 6*sin(t), s], t=-Pi..Pi, s=-8..8),
color=["Green","Red"], style=surface, scaling=constrained, lightmodel=light4);

 


 

Download mw.mw

 

A suggestion for an alternative...

Rouben Rostamian 8646
November 05 2022
1 1

I have not examined the source of the error in your worksheet, but I would suggest using the facilities provided in the Physics[Vectors] package for encoding the Rodrigues formula rather than homogeneous transformation matrices, because the math works more naturally this way.  This worksheet shows how.

> 

restart;
> 

with(Physics[Vectors]):

Rodrigues'  formula:
Rotates the vectorNULL`#mover(mi("v"),mo("→"))` about the unit vecor `#mover(mi("k"),mo("→"))` by the angle alpha:

> 

v__rot_ := v_*cos(alpha) + (k_ &x v_)*sin(alpha) + (k_ . v_)*(1 - cos(alpha))*k_;

v_*cos(alpha)+Physics:-Vectors:-`&x`(k_, v_)*sin(alpha)+Physics:-Vectors:-`.`(k_, v_)*(1-cos(alpha))*k_

Make that into a procedure: Usage:  v__rot = R(`#mover(mi("v"),mo("→"))`, `#mover(mi("k"),mo("→"))`, alpha)

> 

R := unapply(v__rot_, v_, k_, alpha);

proc (v_, k_, alpha) options operator, arrow; v_*cos(alpha)+Physics:-Vectors:-`&x`(k_, v_)*sin(alpha)+Physics:-Vectors:-`.`(k_, v_)*(1-cos(alpha))*k_ end proc

Example 1
 

Pick a unit vector:

> 

1*_i + 2*_j + 3*_k:
k_ := % / sqrt(% . %);

(1/14)*(_i+2*_j+3*_k)*14^(1/2)

Pick an arbitrary vector:

> 

v_ := _i - 3*_j + 2*_k;

_i-3*_j+2*_k

Pick a rotation angle:

> 

alpha := Pi/6;

(1/6)*Pi

Find the rotated vector:

> 

R(v_, k_, alpha):
collect(%, [_i, _j, _k]);

((13/28)*3^(1/2)+(13/28)*14^(1/2)+1/14)*_i+(-(11/7)*3^(1/2)+(1/28)*14^(1/2)+1/7)*_j+((25/28)*3^(1/2)-(5/28)*14^(1/2)+3/14)*_k

> 

 

Example 2

 

Composite rotation

A unit vector:

> 

k__1_ := k_;

(1/14)*(_i+2*_j+3*_k)*14^(1/2)

Another unit vector

> 

k__2_ := subs(_j=-_j, k__1_);

(1/14)*(_i-2*_j+3*_k)*14^(1/2)

Rotate `#mover(mi("v"),mo("→"))` about `#msub(mi("k"),mi("1_"))`by (1/6)*Pi and then rotate the result about `#mi("\`k__2\`")` by (1/4)*Pi:

> 

R(R(v_, k__1_, Pi/6), k__2_, Pi/4):
collect(%, [_i, _j, _k], simplify);

((1/392)*(3*3^(1/2)-4*7^(1/2)+57)*2^(1/2)+(1/392)*(82*7^(1/2)+176)*3^(1/2)+(81/196)*7^(1/2)+3/98)*_i+((1/196)*(-66*3^(1/2)+4*7^(1/2)+174)*2^(1/2)+(1/196)*(7*7^(1/2)-176)*3^(1/2)+(3/196)*7^(1/2)-3/49)*_j+((1/392)*(-89*3^(1/2)-12*7^(1/2)+213)*2^(1/2)+(1/392)*(-18*7^(1/2)+528)*3^(1/2)-(25/196)*7^(1/2)+9/98)*_k

> 

 

> 

 

Download mw.mw

 

 

Solution...

Rouben Rostamian 8646
November 04 2022
1 2
 
> 

restart;
> 

kernelopts(version);

`Maple 2022.2, X86 64 LINUX, Oct 23 2022, Build ID 1657361`

> 

pd1 := diff(u(x,t),t) = diff(u(x,t),x,x) + (1-alpha)*u(x,t);

diff(u(x, t), t) = diff(diff(u(x, t), x), x)+(1-alpha)*u(x, t)

> 

pd2 := diff(v(x,t),t) = mu*diff(v(x,t),x,x) + beta*v(x,t) + alpha*u(x,t);

diff(v(x, t), t) = mu*(diff(diff(v(x, t), x), x))+beta*v(x, t)+alpha*u(x, t)

> 

ics := u(x,0) = Dirac(x), v(x,0) = 0;

u(x, 0) = Dirac(x), v(x, 0) = 0

Maple is unable solve the system sympolically on its own:

> 

pdsolve({pd1, pd2, ics}) assuming t > 0, mu > 0;

but it can, with a little help.

 

The first PDE has only the unknown u, so we solve it for u:

> 

sol1 := pdsolve({pd1,ics[1]}) assuming t > 0;

u(x, t) = (1/2)*exp((1/4)*((-4*alpha+4)*t^2-x^2)/t)/(t^(1/2)*Pi^(1/2))

Then, substitute the result in the second PDE, and solve that for v:

> 

subs(sol1, pd2);
sol2 := pdsolve({%, ics[2]}) assuming t > 0, mu > 0;

diff(v(x, t), t) = mu*(diff(diff(v(x, t), x), x))+beta*v(x, t)+(1/2)*alpha*exp((1/4)*((-4*alpha+4)*t^2-x^2)/t)/(t^(1/2)*Pi^(1/2))

v(x, t) = alpha*((erf((1/2)*(-2*((-alpha-beta+1)/(mu-1))^(1/2)*t+x)/t^(1/2))-erf((1/2)*(-2*t*mu*((-alpha-beta+1)/(mu-1))^(1/2)+x)/(mu^(1/2)*t^(1/2))))*exp((-x*(mu-1)*((-alpha-beta+1)/(mu-1))^(1/2)-((-1+alpha)*mu+beta)*t)/(mu-1))-exp((x*(mu-1)*((-alpha-beta+1)/(mu-1))^(1/2)-((-1+alpha)*mu+beta)*t)/(mu-1))*(erf((1/2)*(2*((-alpha-beta+1)/(mu-1))^(1/2)*t+x)/t^(1/2))-erf((1/2)*(2*t*mu*((-alpha-beta+1)/(mu-1))^(1/2)+x)/(mu^(1/2)*t^(1/2)))))/(((-alpha-beta+1)/(mu-1))^(1/2)*(4*mu-4))

 

Download mw.mw

 

Braces, brackets, and parentheses...

Rouben Rostamian 8646
November 02 2022
2 1

In several places you have used square brackets [...] and curly braces {...} for grouping your mathematical terms.  In Maple, the only delimiters that you can use for that purpose are the parentheses (...).

Adjust your code as necessary and write back if it still does not work.

Simplify...

Rouben Rostamian 8646
October 15 2022
2 1

delvin, you should know that your equations are not comprehensible to people who are unfamiliar with numerals written in the Farsi/Persian script.  If you desire to reach the widest available help, make an effort to write your mathematics in symbols which most of the readers of this forum can understand.

Anyway, here is the answer to your question.

 

> 

restart;

You wish to perform your hand calculation in Maple.  Here is the original expression:

> 

c := (-exp(4*d)*beta^2 + 4*alpha*exp(4*d) + beta^2 - 4*alpha)/(8*exp(2*d));

(1/8)*(-exp(4*d)*beta^2+4*alpha*exp(4*d)+beta^2-4*alpha)/exp(2*d)

and here is the simplified form obtained by Maple:

> 

convert(simplify(c), trigh);

(-(1/4)*beta^2+alpha)*sinh(2*d)
      That agrees with your hand calculation.

 

 

This ought to do it...

Rouben Rostamian 8646
October 13 2022
2 1
 
> 

restart;
> 

g := x -> piecewise(x - 1/3 = 0, 1, 0);

g := proc (x) options operator, arrow; piecewise(x-1/3 = 0, 1, 0) end proc

> 

g(3+1/3);

0

> 

g(0+1/3);

1

 

Download mm.mw

 

 

Solution...

Rouben Rostamian 8646
October 09 2022
2 2
> 

restart;

Take m = 1 and renane "(∂phi)/(∂ xi)" to y:

> 

1/2*y^2 + (1-cos(phi)) = h+2:
solve(%, y):
f := unapply(%[1], phi, h);

proc (phi, h) options operator, arrow; (2*cos(phi)+2*h+2)^(1/2) end proc

> 

seq(f(phi,h), h in [-1,0,1]), seq(-f(phi,h), h in [-1,0,1]);
plot([%], phi=-Pi..3*Pi, size=[800,500]);

2^(1/2)*cos(phi)^(1/2), (2*cos(phi)+2)^(1/2), (2*cos(phi)+4)^(1/2), -2^(1/2)*cos(phi)^(1/2), -(2*cos(phi)+2)^(1/2), -(2*cos(phi)+4)^(1/2)

 

Download mw.mw

 

Infinitely many solutions...

Rouben Rostamian 8646
October 04 2022
3 2

That initial value problem has infinitely many solutions.  Have a look at this worksheet.

> 

restart;
> 

ode := diff(v(t),t) = -2*v(t)^(2/3);

diff(v(t), t) = -2*v(t)^(2/3)

Find the general solution

> 

dsolve(ode);
dsol := isolate(%, v(t));

v(t)^(1/3)+(2/3)*t-c__1 = 0

v(t) = (-(2/3)*t+c__1)^3

Let c__1 = 0:

> 

eval(dsol, c__1 = 0);

v(t) = -(8/27)*t^3

That's a solution of the ODE with the initial condition v(0) = 0.

That, however, is not the only solution.  The following is also

a solution of the initial value problem for any choice of a > 0NULL

> 

sol := piecewise(t < a, 0, -(8*(t-a)^3)/27);

sol := piecewise(t < a, 0, -8*(t-a)^3*(1/27))

Here is what the solution looks like when a = 2:

> 

eval(sol, a=2);
plot(%, t=0..4, color=red, thickness=5);

piecewise(t < 2, 0, -8*(t-2)^3*(1/27))

Download mw.mw

 

 

Assumed variables...

Rouben Rostamian 8646
September 28 2022
0 0

The trailing tilde is not a part of  Omega; it is there to remind you that you have made assumptions on it.

If you don't want to see those reminders, disable them with inserting interface(showassumed=0); near the top of your worksheet.

Convert to D...

Rouben Rostamian 8646
September 24 2022
1 6

mtaylor does not like expressions with "diff" in them, but it works well when converted to "D".  So all you need is:

convert(lhs(Eq1), D):
mtaylor(%, [x,t], 2);

 

Geodesics between two points on a torus...

Rouben Rostamian 8646
September 21 2022
3 7

I am posting this as an "Answer" since the question answered here is quite distinct from the question that was originally asked and answered in this thread.

Geodesics connecting two points on a torus

> 

restart;
> 

with(plots):
> 

with(VariationalCalculus):

The parametric equation of a torus with major and minor radii of a and b.

> 

< (a+b*cos(v))*cos(u), (a+b*cos(v))*sin(u),  b*sin(v) >;
x := unapply(%, u, v):

Vector(3, {(1) = (a+b*cos(v))*cos(u), (2) = (a+b*cos(v))*sin(u), (3) = b*sin(v)})

Calculate the coefficients E, F, and G of the first fundamental form:

> 

xu := diff(x(u,v),u):
xv := diff(x(u,v),v):
E := unapply(simplify(xu^+ . xu), u, v);
F := unapply(simplify(xu^+ . xv), u, v);
G := unapply(simplify(xv^+ . xv), u, v);

proc (u, v) options operator, arrow; (a+b*cos(v))^2 end proc

proc (u, v) options operator, arrow; 0 end proc

proc (u, v) options operator, arrow; b^2 end proc

Look for geodesics of the form x(u, v(u)).  These account for all geodesics with the exception

of those corresponding to "u="constant.  The latter are plain circles so we don't need Maple

to calculate them.

 

From differential geometry, the length of  any curve of the form x(u, v(u)), u__1 < u and u < u__2 

is given by L = int(Upsilon(u), u = u__1 .. u__2) where:

> 

Upsilon := sqrt(E(u, v(u)) + 2*F(u, v(u))*diff(v(u),u) + G(u,v(u))*(diff(v(u),u))^2);

((a+b*cos(v(u)))^2+b^2*(diff(v(u), u))^2)^(1/2)

The minimum length is obtained with the help from the Euler-Lagrange equation

which leads to a second order nonlinear differential equation in v(u):

> 

EulerLagrange(Upsilon, u, v(u)):
remove(type, %, equation)[]:
isolate(%, diff(v(u), u, u)):
de := simplify(%);

diff(diff(v(u), u), u) = -(2*b^2*(diff(v(u), u))^2+(a+b*cos(v(u)))^2)*sin(v(u))/((a+b*cos(v(u)))*b)

Pick numbers for the torus's radii:

> 

a, b := 5, 2;

5, 2

> 

torus := plot3d(x(u,v), u=-Pi..Pi, v=-Pi..Pi,
        scaling=constrained, color="khaki", style=surface);

Example 1

 

Calculate a geodesic from u = 0, v = 0 to u = (1/2)*Pi,  v = (1/2)*Pi:

> 

bc1 := v(0)=0,  v(Pi/2)=Pi/2;

v(0) = 0, v((1/2)*Pi) = (1/2)*Pi

> 

dsol1 := dsolve({de, bc1}, numeric, output=operator);

[u = proc (u) local _res, _dat, _solnproc; option `Copyright (c) 1993 by the University of Waterloo. All rights reserved.`; _dat := Array(1..4, {(1) = proc (outpoint) local X, Y, YP, yout, errproc, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; X := Vector(21, {(1) = .0, (2) = 0.5860933145427747e-1, (3) = .11721866290855494, (4) = .17582799436283242, (5) = .23445728501980884, (6) = .29400251389637233, (7) = .35502246529707177, (8) = .4176324593904259, (9) = .48228601222277023, (10) = .5512774626633876, (11) = .6260306219686177, (12) = .7081249005896944, (13) = .7985125481567577, (14) = .8998553552891154, (15) = 1.0097294694706644, (16) = 1.1235608288918082, (17) = 1.236518680079433, (18) = 1.3442780572143578, (19) = 1.4354896847639433, (20) = 1.5083320801396727, (21) = 1.5707963267949}, datatype = float[8], order = C_order); Y := Matrix(21, 2, {(1, 1) = .0, (1, 2) = 4.357909520225549, (2, 1) = .2533473418001706, (2, 2) = 4.253184219882809, (3, 1) = .4950045392990294, (3, 2) = 3.9682426260835926, (4, 1) = .7163213657367207, (4, 2) = 3.5718304902965725, (5, 1) = .912986520750112, (5, 2) = 3.1352819174892472, (6, 1) = 1.0866622862817927, (6, 2) = 2.7029203986640025, (7, 1) = 1.2390473250060285, (7, 2) = 2.2996023968537105, (8, 1) = 1.3714065523042172, (8, 2) = 1.9372281704148966, (9, 1) = 1.4859944998591417, (9, 2) = 1.6158898922071914, (10, 1) = 1.587137032220874, (10, 2) = 1.3241155689575193, (11, 1) = 1.6758207471908326, (11, 2) = 1.0559025112956482, (12, 1) = 1.7519378227487177, (12, 2) = .8049893321158691, (13, 1) = 1.8136775679360608, (13, 2) = .566571063410118, (14, 1) = 1.8589077269041003, (14, 2) = .33025980575675135, (15, 1) = 1.8821194017143614, (15, 2) = 0.9449920022222437e-1, (16, 1) = 1.879341830166544, (16, 2) = -.1435752401424998, (17, 1) = 1.8494009920056307, (17, 2) = -.3895492919942197, (18, 1) = 1.7937205716265703, (18, 2) = -.649482813775163, (19, 1) = 1.7232026399266667, (19, 2) = -.9032024763979258, (20, 1) = 1.649041988044007, (20, 2) = -1.138744988151322, (21, 1) = 1.5707963267949, (21, 2) = -1.37200094134623}, datatype = float[8], order = C_order); YP := Matrix(21, 2, {(1, 1) = 4.357909520225549, (1, 2) = -.0, (2, 1) = 4.253184219882809, (2, 2) = -3.48400814417279, (3, 1) = 3.9682426260835926, (3, 2) = -6.031898300951247, (4, 1) = 3.5718304902965725, (4, 2) = -7.285198339756213, (5, 1) = 3.1352819174892472, (5, 2) = -7.462358620559608, (6, 1) = 2.7029203986640025, (6, 2) = -6.985675492056729, (7, 1) = 2.2996023968537105, (7, 2) = -6.210452323447571, (8, 1) = 1.9372281704148966, (8, 2) = -5.371378792594195, (9, 1) = 1.6158898922071914, (9, 2) = -4.588579439987948, (10, 1) = 1.3241155689575193, (10, 2) = -3.8949934400656105, (11, 1) = 1.0559025112956482, (11, 2) = -3.3078240571262456, (12, 1) = .8049893321158691, (12, 2) = -2.831415429501529, (13, 1) = .566571063410118, (13, 2) = -2.46897776239132, (14, 1) = .33025980575675135, (14, 2) = -2.2189138501893897, (15, 1) = 0.9449920022222437e-1, (15, 2) = -2.0959796964461206, (16, 1) = -.1435752401424998, (16, 2) = -2.110492861185544, (17, 1) = -.3895492919942197, (17, 2) = -2.2703348506653094, (18, 1) = -.649482813775163, (18, 2) = -2.5835657026971863, (19, 1) = -.9032024763979258, (19, 2) = -3.0077240548867414, (20, 1) = -1.138744988151322, (20, 2) = -3.482022199297429, (21, 1) = -1.37200094134623, (21, 2) = -4.005909266443952}, datatype = float[8], order = C_order); errproc := proc (x_bvp) local outpoint, X, Y, yout, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; Digits := 15; outpoint := evalf(x_bvp); X := Vector(21, {(1) = .0, (2) = 0.5860933145427747e-1, (3) = .11721866290855494, (4) = .17582799436283242, (5) = .23445728501980884, (6) = .29400251389637233, (7) = .35502246529707177, (8) = .4176324593904259, (9) = .48228601222277023, (10) = .5512774626633876, (11) = .6260306219686177, (12) = .7081249005896944, (13) = .7985125481567577, (14) = .8998553552891154, (15) = 1.0097294694706644, (16) = 1.1235608288918082, (17) = 1.236518680079433, (18) = 1.3442780572143578, (19) = 1.4354896847639433, (20) = 1.5083320801396727, (21) = 1.5707963267949}, datatype = float[8], order = C_order); Y := Matrix(21, 2, {(1, 1) = .0, (1, 2) = 0.6072503241630317e-7, (2, 1) = 0.1633853429871433e-7, (2, 2) = -0.17727261949849427e-7, (3, 1) = 0.61119342794151085e-8, (3, 2) = -0.8326391241592253e-7, (4, 1) = -0.8900614449991116e-8, (4, 2) = -0.5046791702922638e-7, (5, 1) = -0.12191319360321298e-7, (5, 2) = -0.2346020405227589e-7, (6, 1) = -0.9575722148775851e-8, (6, 2) = -0.25118370529491878e-7, (7, 1) = -0.7857479569879596e-8, (7, 2) = -0.28341312692007233e-7, (8, 1) = -0.854262175747852e-8, (8, 2) = -0.22349665722449066e-7, (9, 1) = -0.10412904856270964e-7, (9, 2) = -0.12056403907819441e-7, (10, 1) = -0.12496603030736918e-7, (10, 2) = -0.21735212793440864e-8, (11, 1) = -0.14343296103736601e-7, (11, 2) = 0.4884974297168228e-8, (12, 1) = -0.15809361149223226e-7, (12, 2) = 0.8424061973135961e-8, (13, 1) = -0.1684004558944395e-7, (13, 2) = 0.8468555558744448e-8, (14, 1) = -0.17663088227947315e-7, (14, 2) = 0.5752865457923729e-8, (15, 1) = -0.1799958791726186e-7, (15, 2) = 0.8947994861895889e-9, (16, 1) = -0.1754706929898934e-7, (16, 2) = -0.4029590222151902e-8, (17, 1) = -0.15600567545640277e-7, (17, 2) = -0.5508170053250888e-8, (18, 1) = -0.1101866414537828e-7, (18, 2) = 0.17737187382485602e-8, (19, 1) = -0.6481329226736263e-8, (19, 2) = 0.1388172656806454e-7, (20, 1) = -0.31379335721370494e-8, (20, 2) = 0.2492283863873428e-7, (21, 1) = .0, (21, 2) = 0.3612844847603369e-7}, datatype = float[8], order = C_order); if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "right" then return X[21] elif outpoint = "order" then return 6 elif outpoint = "error" then return HFloat(8.326391241592253e-8) elif outpoint = "errorproc" then error "this is already the error procedure" elif outpoint = "rawdata" then return [2, 21, [v(u), diff(v(u), u)], X, Y] else return ('procname')(x_bvp) end if end if; if outpoint < X[1] or X[21] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[21] end if; V := array([1 = 4, 2 = 0]); if Digits <= trunc(evalhf(Digits)) then L := Vector(4, 'datatype' = 'float'[8]); yout := Vector(2, 'datatype' = 'float'[8]); evalhf(`dsolve/numeric/lagrange`(21, 2, X, Y, outpoint, var(yout), var(L), var(V))) else L := Vector(4, 'datatype' = 'sfloat'); yout := Vector(2, 'datatype' = 'sfloat'); `dsolve/numeric/lagrange`(21, 2, X, Y, outpoint, yout, L, V) end if; [u = outpoint, seq('[v(u), diff(v(u), u)]'[i] = yout[i], i = 1 .. 2)] end proc; if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "method" then return "bvp" elif outpoint = "right" then return X[21] elif outpoint = "order" then return 6 elif outpoint = "error" then return HFloat(8.326391241592253e-8) elif outpoint = "errorproc" then return eval(errproc) elif outpoint = "rawdata" then return [2, 21, "depnames", X, Y, YP] else error "non-numeric value" end if end if; if outpoint < X[1] or X[21] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[21] end if; if Digits <= trunc(evalhf(Digits)) and (_EnvInFsolve <> true or _EnvDSNumericSaveDigits <= trunc(evalhf(Digits))) then V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false) ] ); L := Matrix(7, 2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0, (3, 1) = .0, (3, 2) = .0, (4, 1) = .0, (4, 2) = .0, (5, 1) = .0, (5, 2) = .0, (6, 1) = .0, (6, 2) = .0, (7, 1) = .0, (7, 2) = .0}, datatype = float[8], order = C_order); yout := Vector(2, {(1) = .0, (2) = .0}, datatype = float[8]); evalhf(`dsolve/numeric/hermite`(21, 2, X, Y, YP, outpoint, var(yout), var(L), var(V))) else if _EnvInFsolve = true then Digits := _EnvDSNumericSaveDigits end if; V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false) ] ); L := Matrix(7, 2, {(1, 1) = 0., (1, 2) = 0., (2, 1) = 0., (2, 2) = 0., (3, 1) = 0., (3, 2) = 0., (4, 1) = 0., (4, 2) = 0., (5, 1) = 0., (5, 2) = 0., (6, 1) = 0., (6, 2) = 0., (7, 1) = 0., (7, 2) = 0.}, order = C_order); yout := Vector(2, {(1) = 0., (2) = 0.}); `dsolve/numeric/hermite`(21, 2, X, Y, YP, outpoint, yout, L, V) end if; [outpoint, seq(yout[i], i = 1 .. 2)] end proc, (2) = Array(1..3, {(1) = 36893628625364698316, (2) = 36893628625364698756, (3) = 36893628625364698932}), (3) = [u, v(u), diff(v(u), u)], (4) = 0}); _solnproc := _dat[1]; if member(u, ["last", 'last']) then _res := _solnproc("last"); if type(_res, 'list') then return _res[1] end if elif type(u, `=`) and member(lhs(u), ["initial", 'initial']) then if type(rhs(u), 'list') then _res := _solnproc("initial" = [0, op(rhs(u))]) else _res := _solnproc("initial" = [1, rhs(u)]) end if; if type(_res, 'list') then return _res[1] end if elif u = "sysvars" then return _dat[3] end if; u end proc, v = proc (u) local res, data, solnproc, v, outpoint; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then outpoint := evalf[_EnvDSNumericSaveDigits](u) else outpoint := evalf(u) end if; data := Array(1..4, {(1) = proc (outpoint) local X, Y, YP, yout, errproc, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; X := Vector(21, {(1) = .0, (2) = 0.5860933145427747e-1, (3) = .11721866290855494, (4) = .17582799436283242, (5) = .23445728501980884, (6) = .29400251389637233, (7) = .35502246529707177, (8) = .4176324593904259, (9) = .48228601222277023, (10) = .5512774626633876, (11) = .6260306219686177, (12) = .7081249005896944, (13) = .7985125481567577, (14) = .8998553552891154, (15) = 1.0097294694706644, (16) = 1.1235608288918082, (17) = 1.236518680079433, (18) = 1.3442780572143578, (19) = 1.4354896847639433, (20) = 1.5083320801396727, (21) = 1.5707963267949}, datatype = float[8], order = C_order); Y := Matrix(21, 2, {(1, 1) = .0, (1, 2) = 4.357909520225549, (2, 1) = .2533473418001706, (2, 2) = 4.253184219882809, (3, 1) = .4950045392990294, (3, 2) = 3.9682426260835926, (4, 1) = .7163213657367207, (4, 2) = 3.5718304902965725, (5, 1) = .912986520750112, (5, 2) = 3.1352819174892472, (6, 1) = 1.0866622862817927, (6, 2) = 2.7029203986640025, (7, 1) = 1.2390473250060285, (7, 2) = 2.2996023968537105, (8, 1) = 1.3714065523042172, (8, 2) = 1.9372281704148966, (9, 1) = 1.4859944998591417, (9, 2) = 1.6158898922071914, (10, 1) = 1.587137032220874, (10, 2) = 1.3241155689575193, (11, 1) = 1.6758207471908326, (11, 2) = 1.0559025112956482, (12, 1) = 1.7519378227487177, (12, 2) = .8049893321158691, (13, 1) = 1.8136775679360608, (13, 2) = .566571063410118, (14, 1) = 1.8589077269041003, (14, 2) = .33025980575675135, (15, 1) = 1.8821194017143614, (15, 2) = 0.9449920022222437e-1, (16, 1) = 1.879341830166544, (16, 2) = -.1435752401424998, (17, 1) = 1.8494009920056307, (17, 2) = -.3895492919942197, (18, 1) = 1.7937205716265703, (18, 2) = -.649482813775163, (19, 1) = 1.7232026399266667, (19, 2) = -.9032024763979258, (20, 1) = 1.649041988044007, (20, 2) = -1.138744988151322, (21, 1) = 1.5707963267949, (21, 2) = -1.37200094134623}, datatype = float[8], order = C_order); YP := Matrix(21, 2, {(1, 1) = 4.357909520225549, (1, 2) = -.0, (2, 1) = 4.253184219882809, (2, 2) = -3.48400814417279, (3, 1) = 3.9682426260835926, (3, 2) = -6.031898300951247, (4, 1) = 3.5718304902965725, (4, 2) = -7.285198339756213, (5, 1) = 3.1352819174892472, (5, 2) = -7.462358620559608, (6, 1) = 2.7029203986640025, (6, 2) = -6.985675492056729, (7, 1) = 2.2996023968537105, (7, 2) = -6.210452323447571, (8, 1) = 1.9372281704148966, (8, 2) = -5.371378792594195, (9, 1) = 1.6158898922071914, (9, 2) = -4.588579439987948, (10, 1) = 1.3241155689575193, (10, 2) = -3.8949934400656105, (11, 1) = 1.0559025112956482, (11, 2) = -3.3078240571262456, (12, 1) = .8049893321158691, (12, 2) = -2.831415429501529, (13, 1) = .566571063410118, (13, 2) = -2.46897776239132, (14, 1) = .33025980575675135, (14, 2) = -2.2189138501893897, (15, 1) = 0.9449920022222437e-1, (15, 2) = -2.0959796964461206, (16, 1) = -.1435752401424998, (16, 2) = -2.110492861185544, (17, 1) = -.3895492919942197, (17, 2) = -2.2703348506653094, (18, 1) = -.649482813775163, (18, 2) = -2.5835657026971863, (19, 1) = -.9032024763979258, (19, 2) = -3.0077240548867414, (20, 1) = -1.138744988151322, (20, 2) = -3.482022199297429, (21, 1) = -1.37200094134623, (21, 2) = -4.005909266443952}, datatype = float[8], order = C_order); errproc := proc (x_bvp) local outpoint, X, Y, yout, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; Digits := 15; outpoint := evalf(x_bvp); X := Vector(21, {(1) = .0, (2) = 0.5860933145427747e-1, (3) = .11721866290855494, (4) = .17582799436283242, (5) = .23445728501980884, (6) = .29400251389637233, (7) = .35502246529707177, (8) = .4176324593904259, (9) = .48228601222277023, (10) = .5512774626633876, (11) = .6260306219686177, (12) = .7081249005896944, (13) = .7985125481567577, (14) = .8998553552891154, (15) = 1.0097294694706644, (16) = 1.1235608288918082, (17) = 1.236518680079433, (18) = 1.3442780572143578, (19) = 1.4354896847639433, (20) = 1.5083320801396727, (21) = 1.5707963267949}, datatype = float[8], order = C_order); Y := Matrix(21, 2, {(1, 1) = .0, (1, 2) = 0.6072503241630317e-7, (2, 1) = 0.1633853429871433e-7, (2, 2) = -0.17727261949849427e-7, (3, 1) = 0.61119342794151085e-8, (3, 2) = -0.8326391241592253e-7, (4, 1) = -0.8900614449991116e-8, (4, 2) = -0.5046791702922638e-7, (5, 1) = -0.12191319360321298e-7, (5, 2) = -0.2346020405227589e-7, (6, 1) = -0.9575722148775851e-8, (6, 2) = -0.25118370529491878e-7, (7, 1) = -0.7857479569879596e-8, (7, 2) = -0.28341312692007233e-7, (8, 1) = -0.854262175747852e-8, (8, 2) = -0.22349665722449066e-7, (9, 1) = -0.10412904856270964e-7, (9, 2) = -0.12056403907819441e-7, (10, 1) = -0.12496603030736918e-7, (10, 2) = -0.21735212793440864e-8, (11, 1) = -0.14343296103736601e-7, (11, 2) = 0.4884974297168228e-8, (12, 1) = -0.15809361149223226e-7, (12, 2) = 0.8424061973135961e-8, (13, 1) = -0.1684004558944395e-7, (13, 2) = 0.8468555558744448e-8, (14, 1) = -0.17663088227947315e-7, (14, 2) = 0.5752865457923729e-8, (15, 1) = -0.1799958791726186e-7, (15, 2) = 0.8947994861895889e-9, (16, 1) = -0.1754706929898934e-7, (16, 2) = -0.4029590222151902e-8, (17, 1) = -0.15600567545640277e-7, (17, 2) = -0.5508170053250888e-8, (18, 1) = -0.1101866414537828e-7, (18, 2) = 0.17737187382485602e-8, (19, 1) = -0.6481329226736263e-8, (19, 2) = 0.1388172656806454e-7, (20, 1) = -0.31379335721370494e-8, (20, 2) = 0.2492283863873428e-7, (21, 1) = .0, (21, 2) = 0.3612844847603369e-7}, datatype = float[8], order = C_order); if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "right" then return X[21] elif outpoint = "order" then return 6 elif outpoint = "error" then return HFloat(8.326391241592253e-8) elif outpoint = "errorproc" then error "this is already the error procedure" elif outpoint = "rawdata" then return [2, 21, [v(u), diff(v(u), u)], X, Y] else return ('procname')(x_bvp) end if end if; if outpoint < X[1] or X[21] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[21] end if; V := array([1 = 4, 2 = 0]); if Digits <= trunc(evalhf(Digits)) then L := Vector(4, 'datatype' = 'float'[8]); yout := Vector(2, 'datatype' = 'float'[8]); evalhf(`dsolve/numeric/lagrange`(21, 2, X, Y, outpoint, var(yout), var(L), var(V))) else L := Vector(4, 'datatype' = 'sfloat'); yout := Vector(2, 'datatype' = 'sfloat'); `dsolve/numeric/lagrange`(21, 2, X, Y, outpoint, yout, L, V) end if; [u = outpoint, seq('[v(u), diff(v(u), u)]'[i] = yout[i], i = 1 .. 2)] end proc; if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "method" then return "bvp" elif outpoint = "right" then return X[21] elif outpoint = "order" then return 6 elif outpoint = "error" then return HFloat(8.326391241592253e-8) elif outpoint = "errorproc" then return eval(errproc) elif outpoint = "rawdata" then return [2, 21, "depnames", X, Y, YP] else error "non-numeric value" end if end if; if outpoint < X[1] or X[21] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[21] end if; if Digits <= trunc(evalhf(Digits)) and (_EnvInFsolve <> true or _EnvDSNumericSaveDigits <= trunc(evalhf(Digits))) then V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false) ] ); L := Matrix(7, 2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0, (3, 1) = .0, (3, 2) = .0, (4, 1) = .0, (4, 2) = .0, (5, 1) = .0, (5, 2) = .0, (6, 1) = .0, (6, 2) = .0, (7, 1) = .0, (7, 2) = .0}, datatype = float[8], order = C_order); yout := Vector(2, {(1) = .0, (2) = .0}, datatype = float[8]); evalhf(`dsolve/numeric/hermite`(21, 2, X, Y, YP, outpoint, var(yout), var(L), var(V))) else if _EnvInFsolve = true then Digits := _EnvDSNumericSaveDigits end if; V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false) ] ); L := Matrix(7, 2, {(1, 1) = 0., (1, 2) = 0., (2, 1) = 0., (2, 2) = 0., (3, 1) = 0., (3, 2) = 0., (4, 1) = 0., (4, 2) = 0., (5, 1) = 0., (5, 2) = 0., (6, 1) = 0., (6, 2) = 0., (7, 1) = 0., (7, 2) = 0.}, order = C_order); yout := Vector(2, {(1) = 0., (2) = 0.}); `dsolve/numeric/hermite`(21, 2, X, Y, YP, outpoint, yout, L, V) end if; [outpoint, seq(yout[i], i = 1 .. 2)] end proc, (2) = Array(1..3, {(1) = 36893628625364698316, (2) = 36893628625364698756, (3) = 36893628625364698932}), (3) = [u, v(u), diff(v(u), u)], (4) = 0}); solnproc := data[1]; if not type(outpoint, 'numeric') then if outpoint = "solnprocedure" then return eval(solnproc) elif member(outpoint, ["start", "left", "right", "errorproc", "rawdata", "order", "error"]) then return solnproc(u) elif outpoint = "sysvars" then return data[3] elif procname <> unknown then return ('procname')(u) else v := pointto(data[2][2]); return ('v')(u) end if end if; try res := solnproc(outpoint); res[2] catch: error end try end proc, D(v) = proc (u) local res, data, solnproc, `D(v)`, outpoint; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then outpoint := evalf[_EnvDSNumericSaveDigits](u) else outpoint := evalf(u) end if; data := Array(1..4, {(1) = proc (outpoint) local X, Y, YP, yout, errproc, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; X := Vector(21, {(1) = .0, (2) = 0.5860933145427747e-1, (3) = .11721866290855494, (4) = .17582799436283242, (5) = .23445728501980884, (6) = .29400251389637233, (7) = .35502246529707177, (8) = .4176324593904259, (9) = .48228601222277023, (10) = .5512774626633876, (11) = .6260306219686177, (12) = .7081249005896944, (13) = .7985125481567577, (14) = .8998553552891154, (15) = 1.0097294694706644, (16) = 1.1235608288918082, (17) = 1.236518680079433, (18) = 1.3442780572143578, (19) = 1.4354896847639433, (20) = 1.5083320801396727, (21) = 1.5707963267949}, datatype = float[8], order = C_order); Y := Matrix(21, 2, {(1, 1) = .0, (1, 2) = 4.357909520225549, (2, 1) = .2533473418001706, (2, 2) = 4.253184219882809, (3, 1) = .4950045392990294, (3, 2) = 3.9682426260835926, (4, 1) = .7163213657367207, (4, 2) = 3.5718304902965725, (5, 1) = .912986520750112, (5, 2) = 3.1352819174892472, (6, 1) = 1.0866622862817927, (6, 2) = 2.7029203986640025, (7, 1) = 1.2390473250060285, (7, 2) = 2.2996023968537105, (8, 1) = 1.3714065523042172, (8, 2) = 1.9372281704148966, (9, 1) = 1.4859944998591417, (9, 2) = 1.6158898922071914, (10, 1) = 1.587137032220874, (10, 2) = 1.3241155689575193, (11, 1) = 1.6758207471908326, (11, 2) = 1.0559025112956482, (12, 1) = 1.7519378227487177, (12, 2) = .8049893321158691, (13, 1) = 1.8136775679360608, (13, 2) = .566571063410118, (14, 1) = 1.8589077269041003, (14, 2) = .33025980575675135, (15, 1) = 1.8821194017143614, (15, 2) = 0.9449920022222437e-1, (16, 1) = 1.879341830166544, (16, 2) = -.1435752401424998, (17, 1) = 1.8494009920056307, (17, 2) = -.3895492919942197, (18, 1) = 1.7937205716265703, (18, 2) = -.649482813775163, (19, 1) = 1.7232026399266667, (19, 2) = -.9032024763979258, (20, 1) = 1.649041988044007, (20, 2) = -1.138744988151322, (21, 1) = 1.5707963267949, (21, 2) = -1.37200094134623}, datatype = float[8], order = C_order); YP := Matrix(21, 2, {(1, 1) = 4.357909520225549, (1, 2) = -.0, (2, 1) = 4.253184219882809, (2, 2) = -3.48400814417279, (3, 1) = 3.9682426260835926, (3, 2) = -6.031898300951247, (4, 1) = 3.5718304902965725, (4, 2) = -7.285198339756213, (5, 1) = 3.1352819174892472, (5, 2) = -7.462358620559608, (6, 1) = 2.7029203986640025, (6, 2) = -6.985675492056729, (7, 1) = 2.2996023968537105, (7, 2) = -6.210452323447571, (8, 1) = 1.9372281704148966, (8, 2) = -5.371378792594195, (9, 1) = 1.6158898922071914, (9, 2) = -4.588579439987948, (10, 1) = 1.3241155689575193, (10, 2) = -3.8949934400656105, (11, 1) = 1.0559025112956482, (11, 2) = -3.3078240571262456, (12, 1) = .8049893321158691, (12, 2) = -2.831415429501529, (13, 1) = .566571063410118, (13, 2) = -2.46897776239132, (14, 1) = .33025980575675135, (14, 2) = -2.2189138501893897, (15, 1) = 0.9449920022222437e-1, (15, 2) = -2.0959796964461206, (16, 1) = -.1435752401424998, (16, 2) = -2.110492861185544, (17, 1) = -.3895492919942197, (17, 2) = -2.2703348506653094, (18, 1) = -.649482813775163, (18, 2) = -2.5835657026971863, (19, 1) = -.9032024763979258, (19, 2) = -3.0077240548867414, (20, 1) = -1.138744988151322, (20, 2) = -3.482022199297429, (21, 1) = -1.37200094134623, (21, 2) = -4.005909266443952}, datatype = float[8], order = C_order); errproc := proc (x_bvp) local outpoint, X, Y, yout, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; Digits := 15; outpoint := evalf(x_bvp); X := Vector(21, {(1) = .0, (2) = 0.5860933145427747e-1, (3) = .11721866290855494, (4) = .17582799436283242, (5) = .23445728501980884, (6) = .29400251389637233, (7) = .35502246529707177, (8) = .4176324593904259, (9) = .48228601222277023, (10) = .5512774626633876, (11) = .6260306219686177, (12) = .7081249005896944, (13) = .7985125481567577, (14) = .8998553552891154, (15) = 1.0097294694706644, (16) = 1.1235608288918082, (17) = 1.236518680079433, (18) = 1.3442780572143578, (19) = 1.4354896847639433, (20) = 1.5083320801396727, (21) = 1.5707963267949}, datatype = float[8], order = C_order); Y := Matrix(21, 2, {(1, 1) = .0, (1, 2) = 0.6072503241630317e-7, (2, 1) = 0.1633853429871433e-7, (2, 2) = -0.17727261949849427e-7, (3, 1) = 0.61119342794151085e-8, (3, 2) = -0.8326391241592253e-7, (4, 1) = -0.8900614449991116e-8, (4, 2) = -0.5046791702922638e-7, (5, 1) = -0.12191319360321298e-7, (5, 2) = -0.2346020405227589e-7, (6, 1) = -0.9575722148775851e-8, (6, 2) = -0.25118370529491878e-7, (7, 1) = -0.7857479569879596e-8, (7, 2) = -0.28341312692007233e-7, (8, 1) = -0.854262175747852e-8, (8, 2) = -0.22349665722449066e-7, (9, 1) = -0.10412904856270964e-7, (9, 2) = -0.12056403907819441e-7, (10, 1) = -0.12496603030736918e-7, (10, 2) = -0.21735212793440864e-8, (11, 1) = -0.14343296103736601e-7, (11, 2) = 0.4884974297168228e-8, (12, 1) = -0.15809361149223226e-7, (12, 2) = 0.8424061973135961e-8, (13, 1) = -0.1684004558944395e-7, (13, 2) = 0.8468555558744448e-8, (14, 1) = -0.17663088227947315e-7, (14, 2) = 0.5752865457923729e-8, (15, 1) = -0.1799958791726186e-7, (15, 2) = 0.8947994861895889e-9, (16, 1) = -0.1754706929898934e-7, (16, 2) = -0.4029590222151902e-8, (17, 1) = -0.15600567545640277e-7, (17, 2) = -0.5508170053250888e-8, (18, 1) = -0.1101866414537828e-7, (18, 2) = 0.17737187382485602e-8, (19, 1) = -0.6481329226736263e-8, (19, 2) = 0.1388172656806454e-7, (20, 1) = -0.31379335721370494e-8, (20, 2) = 0.2492283863873428e-7, (21, 1) = .0, (21, 2) = 0.3612844847603369e-7}, datatype = float[8], order = C_order); if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "right" then return X[21] elif outpoint = "order" then return 6 elif outpoint = "error" then return HFloat(8.326391241592253e-8) elif outpoint = "errorproc" then error "this is already the error procedure" elif outpoint = "rawdata" then return [2, 21, [v(u), diff(v(u), u)], X, Y] else return ('procname')(x_bvp) end if end if; if outpoint < X[1] or X[21] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[21] end if; V := array([1 = 4, 2 = 0]); if Digits <= trunc(evalhf(Digits)) then L := Vector(4, 'datatype' = 'float'[8]); yout := Vector(2, 'datatype' = 'float'[8]); evalhf(`dsolve/numeric/lagrange`(21, 2, X, Y, outpoint, var(yout), var(L), var(V))) else L := Vector(4, 'datatype' = 'sfloat'); yout := Vector(2, 'datatype' = 'sfloat'); `dsolve/numeric/lagrange`(21, 2, X, Y, outpoint, yout, L, V) end if; [u = outpoint, seq('[v(u), diff(v(u), u)]'[i] = yout[i], i = 1 .. 2)] end proc; if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "method" then return "bvp" elif outpoint = "right" then return X[21] elif outpoint = "order" then return 6 elif outpoint = "error" then return HFloat(8.326391241592253e-8) elif outpoint = "errorproc" then return eval(errproc) elif outpoint = "rawdata" then return [2, 21, "depnames", X, Y, YP] else error "non-numeric value" end if end if; if outpoint < X[1] or X[21] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[21] end if; if Digits <= trunc(evalhf(Digits)) and (_EnvInFsolve <> true or _EnvDSNumericSaveDigits <= trunc(evalhf(Digits))) then V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false) ] ); L := Matrix(7, 2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0, (3, 1) = .0, (3, 2) = .0, (4, 1) = .0, (4, 2) = .0, (5, 1) = .0, (5, 2) = .0, (6, 1) = .0, (6, 2) = .0, (7, 1) = .0, (7, 2) = .0}, datatype = float[8], order = C_order); yout := Vector(2, {(1) = .0, (2) = .0}, datatype = float[8]); evalhf(`dsolve/numeric/hermite`(21, 2, X, Y, YP, outpoint, var(yout), var(L), var(V))) else if _EnvInFsolve = true then Digits := _EnvDSNumericSaveDigits end if; V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false) ] ); L := Matrix(7, 2, {(1, 1) = 0., (1, 2) = 0., (2, 1) = 0., (2, 2) = 0., (3, 1) = 0., (3, 2) = 0., (4, 1) = 0., (4, 2) = 0., (5, 1) = 0., (5, 2) = 0., (6, 1) = 0., (6, 2) = 0., (7, 1) = 0., (7, 2) = 0.}, order = C_order); yout := Vector(2, {(1) = 0., (2) = 0.}); `dsolve/numeric/hermite`(21, 2, X, Y, YP, outpoint, yout, L, V) end if; [outpoint, seq(yout[i], i = 1 .. 2)] end proc, (2) = Array(1..3, {(1) = 36893628625364698316, (2) = 36893628625364698756, (3) = 36893628625364698932}), (3) = [u, v(u), diff(v(u), u)], (4) = 0}); solnproc := data[1]; if not type(outpoint, 'numeric') then if outpoint = "solnprocedure" then return eval(solnproc) elif member(outpoint, ["start", "left", "right", "errorproc", "rawdata", "order", "error"]) then return solnproc(u) elif outpoint = "sysvars" then return data[3] elif procname <> unknown then return ('procname')(u) else `D(v)` := pointto(data[2][3]); return ('`D(v)`')(u) end if end if; try res := solnproc(outpoint); res[3] catch: error end try end proc]

> 

display(
        torus,
        tubeplot([seq(eval(x(u, v(u)), dsol1))], u=0..Pi/2, color=red, radius=0.1),
        pointplot3d(eval([x(0,v(0)),x(Pi/2,v(Pi/2))], dsol1), symbol=solidsphere, symbolsize=20, color="Orange"),
lightmodel=light4, thickness=5, orientation=[-5,75,0], size=[700,400],
style=surface, axes=none);

> 

 

Example 2

 
> 

bc2 := v(0)=0,  v(Pi/2)=Pi;

v(0) = 0, v((1/2)*Pi) = Pi

> 

dsol2 := dsolve({de, bc2}, numeric, output=operator);

[u = proc (u) local _res, _dat, _solnproc; option `Copyright (c) 1993 by the University of Waterloo. All rights reserved.`; _dat := Array(1..4, {(1) = proc (outpoint) local X, Y, YP, yout, errproc, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; X := Vector(25, {(1) = .0, (2) = 0.3556697983665059e-1, (3) = 0.7150815650209673e-1, (4) = .1078355952208751, (5) = .14456202386531472, (6) = .18170088505495013, (7) = .2192663936438809, (8) = .25772582968519775, (9) = .2979698369946872, (10) = .34028667559059855, (11) = .38503401227598166, (12) = .43268598844062967, (13) = .48418267682069066, (14) = .54178398151824, (15) = .6084908129577877, (16) = .6905705623174822, (17) = .7936582413926776, (18) = .9362490342852545, (19) = 1.1290955386275305, (20) = 1.2785289405368818, (21) = 1.3762683057594027, (22) = 1.4410821559755675, (23) = 1.4911439441371346, (24) = 1.5334604576293163, (25) = 1.5707963267949}, datatype = float[8], order = C_order); Y := Matrix(25, 2, {(1, 1) = .0, (1, 2) = 8.027222261980723, (2, 1) = .2831190340450847, (2, 2) = 7.828369059489753, (3, 1) = .5555864658560054, (3, 2) = 7.287670355233425, (4, 1) = .8071856825371913, (4, 2) = 6.54481306582631, (5, 1) = 1.0328331222442264, (5, 2) = 5.745156668353998, (6, 1) = 1.2318715807045222, (6, 2) = 4.987033344028576, (7, 1) = 1.4062899344379456, (7, 2) = 4.316682182254274, (8, 1) = 1.5608590578898303, (8, 2) = 3.739193866869814, (9, 1) = 1.7009934741789976, (9, 2) = 3.241862801797417, (10, 1) = 1.8288761198346395, (10, 2) = 2.817246508505084, (11, 1) = 1.9465497589776795, (11, 2) = 2.455373052357135, (12, 1) = 2.0559118912274252, (12, 2) = 2.146059224600939, (13, 1) = 2.1592833618063074, (13, 2) = 1.8786440703412244, (14, 1) = 2.260381800745741, (14, 2) = 1.6409272186274388, (15, 1) = 2.362338845054942, (15, 2) = 1.4249780044463283, (16, 1) = 2.4705572534383973, (16, 2) = 1.2215356731044336, (17, 1) = 2.5862348892169664, (17, 2) = 1.0329309045427477, (18, 1) = 2.7197502962411266, (18, 2) = .8522354096629391, (19, 1) = 2.868097308964122, (19, 2) = .7002166019075773, (20, 1) = 2.9670989492898685, (20, 2) = .6306046195751053, (21, 1) = 3.0272505993270076, (21, 2) = .6023260373483883, (22, 1) = 3.065868563957721, (22, 2) = .5901748294236376, (23, 1) = 3.095251805141905, (23, 2) = .5841893523046803, (24, 1) = 3.11990600006039, (24, 2) = .5813793052108006, (25, 1) = 3.14159265358979, (25, 2) = .5805900104831727}, datatype = float[8], order = C_order); YP := Matrix(25, 2, {(1, 1) = 8.027222261980723, (1, 2) = -.0, (2, 1) = 7.828369059489753, (2, 2) = -10.861805181334308, (3, 1) = 7.287670355233425, (3, 2) = -18.492634875073744, (4, 1) = 6.54481306582631, (4, 2) = -21.694990694475745, (5, 1) = 5.745156668353998, (5, 2) = -21.40568916672294, (6, 1) = 4.987033344028576, (6, 2) = -19.233324103734176, (7, 1) = 4.316682182254274, (7, 2) = -16.42945105476428, (8, 1) = 3.739193866869814, (8, 2) = -13.650236112103359, (9, 1) = 3.241862801797417, (9, 2) = -11.143337685208598, (10, 1) = 2.817246508505084, (10, 2) = -9.007662429827185, (11, 1) = 2.455373052357135, (11, 2) = -7.242681944771625, (12, 1) = 2.146059224600939, (12, 2) = -5.805737354677115, (13, 1) = 1.8786440703412244, (13, 2) = -4.636510741573245, (14, 1) = 1.6409272186274388, (14, 2) = -3.6671563351547207, (15, 1) = 1.4249780044463283, (15, 2) = -2.852577215638926, (16, 1) = 1.2215356731044336, (16, 2) = -2.1483661147204827, (17, 1) = 1.0329309045427477, (17, 2) = -1.5518631600869446, (18, 1) = .8522354096629391, (18, 2) = -1.024669368486231, (19, 1) = .7002166019075773, (19, 2) = -.5874907987071298, (20, 1) = .6306046195751053, (20, 2) = -.35417872498610686, (21, 1) = .6023260373483883, (21, 2) = -.22683550784375403, (22, 1) = .5901748294236376, (22, 2) = -.14876072185358538, (23, 1) = .5841893523046803, (23, 2) = -0.9060027226914852e-1, (24, 1) = .5813793052108006, (24, 2) = -0.4230373846316707e-1, (25, 1) = .5805900104831727, (25, 2) = -0.6298834938184995e-14}, datatype = float[8], order = C_order); errproc := proc (x_bvp) local outpoint, X, Y, yout, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; Digits := 15; outpoint := evalf(x_bvp); X := Vector(25, {(1) = .0, (2) = 0.3556697983665059e-1, (3) = 0.7150815650209673e-1, (4) = .1078355952208751, (5) = .14456202386531472, (6) = .18170088505495013, (7) = .2192663936438809, (8) = .25772582968519775, (9) = .2979698369946872, (10) = .34028667559059855, (11) = .38503401227598166, (12) = .43268598844062967, (13) = .48418267682069066, (14) = .54178398151824, (15) = .6084908129577877, (16) = .6905705623174822, (17) = .7936582413926776, (18) = .9362490342852545, (19) = 1.1290955386275305, (20) = 1.2785289405368818, (21) = 1.3762683057594027, (22) = 1.4410821559755675, (23) = 1.4911439441371346, (24) = 1.5334604576293163, (25) = 1.5707963267949}, datatype = float[8], order = C_order); Y := Matrix(25, 2, {(1, 1) = .0, (1, 2) = 0.31500831778913094e-6, (2, 1) = 0.3132220020535465e-7, (2, 2) = 0.7779131439750735e-7, (3, 1) = 0.14325613236197187e-7, (3, 2) = -0.6898540876289522e-7, (4, 1) = -0.6036355453398498e-8, (4, 2) = 0.7943001698758059e-7, (5, 1) = -0.13260007889026392e-8, (5, 2) = 0.11756415179265621e-6, (6, 1) = 0.1071471554168767e-7, (6, 2) = 0.44460301809323215e-7, (7, 1) = 0.16948724685756332e-7, (7, 2) = -0.6994714895790386e-8, (8, 1) = 0.1782055631368677e-7, (8, 2) = -0.1641681279404671e-7, (9, 1) = 0.16393950975769278e-7, (9, 2) = -0.58427989726364065e-8, (10, 1) = 0.14618894539761003e-7, (10, 2) = 0.7611526601257115e-8, (11, 1) = 0.13363200796132462e-7, (11, 2) = 0.16937803808645205e-7, (12, 1) = 0.12801606379592921e-7, (12, 2) = 0.21469959634400602e-7, (13, 1) = 0.12794426509551367e-7, (13, 2) = 0.22808086897112244e-7, (14, 1) = 0.13080732166085607e-7, (14, 2) = 0.22688679611190916e-7, (15, 1) = 0.13483594578904676e-7, (15, 2) = 0.2185857499871233e-7, (16, 1) = 0.13492144691744485e-7, (16, 2) = 0.2072575098426527e-7, (17, 1) = 0.12840479015099201e-7, (17, 2) = 0.17970028302554435e-7, (18, 1) = 0.720289485656478e-8, (18, 2) = 0.11283105136137962e-7, (19, 1) = -0.10626354523447358e-8, (19, 2) = -0.7464659107819674e-8, (20, 1) = 0.19375865427224214e-8, (20, 2) = -0.1047570220567349e-7, (21, 1) = 0.15016966694503165e-8, (21, 2) = -0.914149161086985e-8, (22, 1) = 0.10081859027200914e-8, (22, 2) = -0.848749960193092e-8, (23, 1) = 0.618147262744349e-9, (23, 2) = -0.8172877515951791e-8, (24, 1) = 0.2897545785236149e-9, (24, 2) = -0.8027811845254448e-8, (25, 1) = .0, (25, 2) = -0.7987456326182811e-8}, datatype = float[8], order = C_order); if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "right" then return X[25] elif outpoint = "order" then return 6 elif outpoint = "error" then return HFloat(3.1500831778913094e-7) elif outpoint = "errorproc" then error "this is already the error procedure" elif outpoint = "rawdata" then return [2, 25, [v(u), diff(v(u), u)], X, Y] else return ('procname')(x_bvp) end if end if; if outpoint < X[1] or X[25] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[25] end if; V := array([1 = 4, 2 = 0]); if Digits <= trunc(evalhf(Digits)) then L := Vector(4, 'datatype' = 'float'[8]); yout := Vector(2, 'datatype' = 'float'[8]); evalhf(`dsolve/numeric/lagrange`(25, 2, X, Y, outpoint, var(yout), var(L), var(V))) else L := Vector(4, 'datatype' = 'sfloat'); yout := Vector(2, 'datatype' = 'sfloat'); `dsolve/numeric/lagrange`(25, 2, X, Y, outpoint, yout, L, V) end if; [u = outpoint, seq('[v(u), diff(v(u), u)]'[i] = yout[i], i = 1 .. 2)] end proc; if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "method" then return "bvp" elif outpoint = "right" then return X[25] elif outpoint = "order" then return 6 elif outpoint = "error" then return HFloat(3.1500831778913094e-7) elif outpoint = "errorproc" then return eval(errproc) elif outpoint = "rawdata" then return [2, 25, "depnames", X, Y, YP] else error "non-numeric value" end if end if; if outpoint < X[1] or X[25] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[25] end if; if Digits <= trunc(evalhf(Digits)) and (_EnvInFsolve <> true or _EnvDSNumericSaveDigits <= trunc(evalhf(Digits))) then V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false) ] ); L := Matrix(7, 2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0, (3, 1) = .0, (3, 2) = .0, (4, 1) = .0, (4, 2) = .0, (5, 1) = .0, (5, 2) = .0, (6, 1) = .0, (6, 2) = .0, (7, 1) = .0, (7, 2) = .0}, datatype = float[8], order = C_order); yout := Vector(2, {(1) = .0, (2) = .0}, datatype = float[8]); evalhf(`dsolve/numeric/hermite`(25, 2, X, Y, YP, outpoint, var(yout), var(L), var(V))) else if _EnvInFsolve = true then Digits := _EnvDSNumericSaveDigits end if; V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false) ] ); L := Matrix(7, 2, {(1, 1) = 0., (1, 2) = 0., (2, 1) = 0., (2, 2) = 0., (3, 1) = 0., (3, 2) = 0., (4, 1) = 0., (4, 2) = 0., (5, 1) = 0., (5, 2) = 0., (6, 1) = 0., (6, 2) = 0., (7, 1) = 0., (7, 2) = 0.}, order = C_order); yout := Vector(2, {(1) = 0., (2) = 0.}); `dsolve/numeric/hermite`(25, 2, X, Y, YP, outpoint, yout, L, V) end if; [outpoint, seq(yout[i], i = 1 .. 2)] end proc, (2) = Array(1..3, {(1) = 36893628625364672284, (2) = 36893628625364664316, (3) = 36893628625364664492}), (3) = [u, v(u), diff(v(u), u)], (4) = 0}); _solnproc := _dat[1]; if member(u, ["last", 'last']) then _res := _solnproc("last"); if type(_res, 'list') then return _res[1] end if elif type(u, `=`) and member(lhs(u), ["initial", 'initial']) then if type(rhs(u), 'list') then _res := _solnproc("initial" = [0, op(rhs(u))]) else _res := _solnproc("initial" = [1, rhs(u)]) end if; if type(_res, 'list') then return _res[1] end if elif u = "sysvars" then return _dat[3] end if; u end proc, v = proc (u) local res, data, solnproc, v, outpoint; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then outpoint := evalf[_EnvDSNumericSaveDigits](u) else outpoint := evalf(u) end if; data := Array(1..4, {(1) = proc (outpoint) local X, Y, YP, yout, errproc, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; X := Vector(25, {(1) = .0, (2) = 0.3556697983665059e-1, (3) = 0.7150815650209673e-1, (4) = .1078355952208751, (5) = .14456202386531472, (6) = .18170088505495013, (7) = .2192663936438809, (8) = .25772582968519775, (9) = .2979698369946872, (10) = .34028667559059855, (11) = .38503401227598166, (12) = .43268598844062967, (13) = .48418267682069066, (14) = .54178398151824, (15) = .6084908129577877, (16) = .6905705623174822, (17) = .7936582413926776, (18) = .9362490342852545, (19) = 1.1290955386275305, (20) = 1.2785289405368818, (21) = 1.3762683057594027, (22) = 1.4410821559755675, (23) = 1.4911439441371346, (24) = 1.5334604576293163, (25) = 1.5707963267949}, datatype = float[8], order = C_order); Y := Matrix(25, 2, {(1, 1) = .0, (1, 2) = 8.027222261980723, (2, 1) = .2831190340450847, (2, 2) = 7.828369059489753, (3, 1) = .5555864658560054, (3, 2) = 7.287670355233425, (4, 1) = .8071856825371913, (4, 2) = 6.54481306582631, (5, 1) = 1.0328331222442264, (5, 2) = 5.745156668353998, (6, 1) = 1.2318715807045222, (6, 2) = 4.987033344028576, (7, 1) = 1.4062899344379456, (7, 2) = 4.316682182254274, (8, 1) = 1.5608590578898303, (8, 2) = 3.739193866869814, (9, 1) = 1.7009934741789976, (9, 2) = 3.241862801797417, (10, 1) = 1.8288761198346395, (10, 2) = 2.817246508505084, (11, 1) = 1.9465497589776795, (11, 2) = 2.455373052357135, (12, 1) = 2.0559118912274252, (12, 2) = 2.146059224600939, (13, 1) = 2.1592833618063074, (13, 2) = 1.8786440703412244, (14, 1) = 2.260381800745741, (14, 2) = 1.6409272186274388, (15, 1) = 2.362338845054942, (15, 2) = 1.4249780044463283, (16, 1) = 2.4705572534383973, (16, 2) = 1.2215356731044336, (17, 1) = 2.5862348892169664, (17, 2) = 1.0329309045427477, (18, 1) = 2.7197502962411266, (18, 2) = .8522354096629391, (19, 1) = 2.868097308964122, (19, 2) = .7002166019075773, (20, 1) = 2.9670989492898685, (20, 2) = .6306046195751053, (21, 1) = 3.0272505993270076, (21, 2) = .6023260373483883, (22, 1) = 3.065868563957721, (22, 2) = .5901748294236376, (23, 1) = 3.095251805141905, (23, 2) = .5841893523046803, (24, 1) = 3.11990600006039, (24, 2) = .5813793052108006, (25, 1) = 3.14159265358979, (25, 2) = .5805900104831727}, datatype = float[8], order = C_order); YP := Matrix(25, 2, {(1, 1) = 8.027222261980723, (1, 2) = -.0, (2, 1) = 7.828369059489753, (2, 2) = -10.861805181334308, (3, 1) = 7.287670355233425, (3, 2) = -18.492634875073744, (4, 1) = 6.54481306582631, (4, 2) = -21.694990694475745, (5, 1) = 5.745156668353998, (5, 2) = -21.40568916672294, (6, 1) = 4.987033344028576, (6, 2) = -19.233324103734176, (7, 1) = 4.316682182254274, (7, 2) = -16.42945105476428, (8, 1) = 3.739193866869814, (8, 2) = -13.650236112103359, (9, 1) = 3.241862801797417, (9, 2) = -11.143337685208598, (10, 1) = 2.817246508505084, (10, 2) = -9.007662429827185, (11, 1) = 2.455373052357135, (11, 2) = -7.242681944771625, (12, 1) = 2.146059224600939, (12, 2) = -5.805737354677115, (13, 1) = 1.8786440703412244, (13, 2) = -4.636510741573245, (14, 1) = 1.6409272186274388, (14, 2) = -3.6671563351547207, (15, 1) = 1.4249780044463283, (15, 2) = -2.852577215638926, (16, 1) = 1.2215356731044336, (16, 2) = -2.1483661147204827, (17, 1) = 1.0329309045427477, (17, 2) = -1.5518631600869446, (18, 1) = .8522354096629391, (18, 2) = -1.024669368486231, (19, 1) = .7002166019075773, (19, 2) = -.5874907987071298, (20, 1) = .6306046195751053, (20, 2) = -.35417872498610686, (21, 1) = .6023260373483883, (21, 2) = -.22683550784375403, (22, 1) = .5901748294236376, (22, 2) = -.14876072185358538, (23, 1) = .5841893523046803, (23, 2) = -0.9060027226914852e-1, (24, 1) = .5813793052108006, (24, 2) = -0.4230373846316707e-1, (25, 1) = .5805900104831727, (25, 2) = -0.6298834938184995e-14}, datatype = float[8], order = C_order); errproc := proc (x_bvp) local outpoint, X, Y, yout, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; Digits := 15; outpoint := evalf(x_bvp); X := Vector(25, {(1) = .0, (2) = 0.3556697983665059e-1, (3) = 0.7150815650209673e-1, (4) = .1078355952208751, (5) = .14456202386531472, (6) = .18170088505495013, (7) = .2192663936438809, (8) = .25772582968519775, (9) = .2979698369946872, (10) = .34028667559059855, (11) = .38503401227598166, (12) = .43268598844062967, (13) = .48418267682069066, (14) = .54178398151824, (15) = .6084908129577877, (16) = .6905705623174822, (17) = .7936582413926776, (18) = .9362490342852545, (19) = 1.1290955386275305, (20) = 1.2785289405368818, (21) = 1.3762683057594027, (22) = 1.4410821559755675, (23) = 1.4911439441371346, (24) = 1.5334604576293163, (25) = 1.5707963267949}, datatype = float[8], order = C_order); Y := Matrix(25, 2, {(1, 1) = .0, (1, 2) = 0.31500831778913094e-6, (2, 1) = 0.3132220020535465e-7, (2, 2) = 0.7779131439750735e-7, (3, 1) = 0.14325613236197187e-7, (3, 2) = -0.6898540876289522e-7, (4, 1) = -0.6036355453398498e-8, (4, 2) = 0.7943001698758059e-7, (5, 1) = -0.13260007889026392e-8, (5, 2) = 0.11756415179265621e-6, (6, 1) = 0.1071471554168767e-7, (6, 2) = 0.44460301809323215e-7, (7, 1) = 0.16948724685756332e-7, (7, 2) = -0.6994714895790386e-8, (8, 1) = 0.1782055631368677e-7, (8, 2) = -0.1641681279404671e-7, (9, 1) = 0.16393950975769278e-7, (9, 2) = -0.58427989726364065e-8, (10, 1) = 0.14618894539761003e-7, (10, 2) = 0.7611526601257115e-8, (11, 1) = 0.13363200796132462e-7, (11, 2) = 0.16937803808645205e-7, (12, 1) = 0.12801606379592921e-7, (12, 2) = 0.21469959634400602e-7, (13, 1) = 0.12794426509551367e-7, (13, 2) = 0.22808086897112244e-7, (14, 1) = 0.13080732166085607e-7, (14, 2) = 0.22688679611190916e-7, (15, 1) = 0.13483594578904676e-7, (15, 2) = 0.2185857499871233e-7, (16, 1) = 0.13492144691744485e-7, (16, 2) = 0.2072575098426527e-7, (17, 1) = 0.12840479015099201e-7, (17, 2) = 0.17970028302554435e-7, (18, 1) = 0.720289485656478e-8, (18, 2) = 0.11283105136137962e-7, (19, 1) = -0.10626354523447358e-8, (19, 2) = -0.7464659107819674e-8, (20, 1) = 0.19375865427224214e-8, (20, 2) = -0.1047570220567349e-7, (21, 1) = 0.15016966694503165e-8, (21, 2) = -0.914149161086985e-8, (22, 1) = 0.10081859027200914e-8, (22, 2) = -0.848749960193092e-8, (23, 1) = 0.618147262744349e-9, (23, 2) = -0.8172877515951791e-8, (24, 1) = 0.2897545785236149e-9, (24, 2) = -0.8027811845254448e-8, (25, 1) = .0, (25, 2) = -0.7987456326182811e-8}, datatype = float[8], order = C_order); if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "right" then return X[25] elif outpoint = "order" then return 6 elif outpoint = "error" then return HFloat(3.1500831778913094e-7) elif outpoint = "errorproc" then error "this is already the error procedure" elif outpoint = "rawdata" then return [2, 25, [v(u), diff(v(u), u)], X, Y] else return ('procname')(x_bvp) end if end if; if outpoint < X[1] or X[25] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[25] end if; V := array([1 = 4, 2 = 0]); if Digits <= trunc(evalhf(Digits)) then L := Vector(4, 'datatype' = 'float'[8]); yout := Vector(2, 'datatype' = 'float'[8]); evalhf(`dsolve/numeric/lagrange`(25, 2, X, Y, outpoint, var(yout), var(L), var(V))) else L := Vector(4, 'datatype' = 'sfloat'); yout := Vector(2, 'datatype' = 'sfloat'); `dsolve/numeric/lagrange`(25, 2, X, Y, outpoint, yout, L, V) end if; [u = outpoint, seq('[v(u), diff(v(u), u)]'[i] = yout[i], i = 1 .. 2)] end proc; if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "method" then return "bvp" elif outpoint = "right" then return X[25] elif outpoint = "order" then return 6 elif outpoint = "error" then return HFloat(3.1500831778913094e-7) elif outpoint = "errorproc" then return eval(errproc) elif outpoint = "rawdata" then return [2, 25, "depnames", X, Y, YP] else error "non-numeric value" end if end if; if outpoint < X[1] or X[25] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[25] end if; if Digits <= trunc(evalhf(Digits)) and (_EnvInFsolve <> true or _EnvDSNumericSaveDigits <= trunc(evalhf(Digits))) then V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false) ] ); L := Matrix(7, 2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0, (3, 1) = .0, (3, 2) = .0, (4, 1) = .0, (4, 2) = .0, (5, 1) = .0, (5, 2) = .0, (6, 1) = .0, (6, 2) = .0, (7, 1) = .0, (7, 2) = .0}, datatype = float[8], order = C_order); yout := Vector(2, {(1) = .0, (2) = .0}, datatype = float[8]); evalhf(`dsolve/numeric/hermite`(25, 2, X, Y, YP, outpoint, var(yout), var(L), var(V))) else if _EnvInFsolve = true then Digits := _EnvDSNumericSaveDigits end if; V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false) ] ); L := Matrix(7, 2, {(1, 1) = 0., (1, 2) = 0., (2, 1) = 0., (2, 2) = 0., (3, 1) = 0., (3, 2) = 0., (4, 1) = 0., (4, 2) = 0., (5, 1) = 0., (5, 2) = 0., (6, 1) = 0., (6, 2) = 0., (7, 1) = 0., (7, 2) = 0.}, order = C_order); yout := Vector(2, {(1) = 0., (2) = 0.}); `dsolve/numeric/hermite`(25, 2, X, Y, YP, outpoint, yout, L, V) end if; [outpoint, seq(yout[i], i = 1 .. 2)] end proc, (2) = Array(1..3, {(1) = 36893628625364672284, (2) = 36893628625364664316, (3) = 36893628625364664492}), (3) = [u, v(u), diff(v(u), u)], (4) = 0}); solnproc := data[1]; if not type(outpoint, 'numeric') then if outpoint = "solnprocedure" then return eval(solnproc) elif member(outpoint, ["start", "left", "right", "errorproc", "rawdata", "order", "error"]) then return solnproc(u) elif outpoint = "sysvars" then return data[3] elif procname <> unknown then return ('procname')(u) else v := pointto(data[2][2]); return ('v')(u) end if end if; try res := solnproc(outpoint); res[2] catch: error end try end proc, D(v) = proc (u) local res, data, solnproc, `D(v)`, outpoint; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then outpoint := evalf[_EnvDSNumericSaveDigits](u) else outpoint := evalf(u) end if; data := Array(1..4, {(1) = proc (outpoint) local X, Y, YP, yout, errproc, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; X := Vector(25, {(1) = .0, (2) = 0.3556697983665059e-1, (3) = 0.7150815650209673e-1, (4) = .1078355952208751, (5) = .14456202386531472, (6) = .18170088505495013, (7) = .2192663936438809, (8) = .25772582968519775, (9) = .2979698369946872, (10) = .34028667559059855, (11) = .38503401227598166, (12) = .43268598844062967, (13) = .48418267682069066, (14) = .54178398151824, (15) = .6084908129577877, (16) = .6905705623174822, (17) = .7936582413926776, (18) = .9362490342852545, (19) = 1.1290955386275305, (20) = 1.2785289405368818, (21) = 1.3762683057594027, (22) = 1.4410821559755675, (23) = 1.4911439441371346, (24) = 1.5334604576293163, (25) = 1.5707963267949}, datatype = float[8], order = C_order); Y := Matrix(25, 2, {(1, 1) = .0, (1, 2) = 8.027222261980723, (2, 1) = .2831190340450847, (2, 2) = 7.828369059489753, (3, 1) = .5555864658560054, (3, 2) = 7.287670355233425, (4, 1) = .8071856825371913, (4, 2) = 6.54481306582631, (5, 1) = 1.0328331222442264, (5, 2) = 5.745156668353998, (6, 1) = 1.2318715807045222, (6, 2) = 4.987033344028576, (7, 1) = 1.4062899344379456, (7, 2) = 4.316682182254274, (8, 1) = 1.5608590578898303, (8, 2) = 3.739193866869814, (9, 1) = 1.7009934741789976, (9, 2) = 3.241862801797417, (10, 1) = 1.8288761198346395, (10, 2) = 2.817246508505084, (11, 1) = 1.9465497589776795, (11, 2) = 2.455373052357135, (12, 1) = 2.0559118912274252, (12, 2) = 2.146059224600939, (13, 1) = 2.1592833618063074, (13, 2) = 1.8786440703412244, (14, 1) = 2.260381800745741, (14, 2) = 1.6409272186274388, (15, 1) = 2.362338845054942, (15, 2) = 1.4249780044463283, (16, 1) = 2.4705572534383973, (16, 2) = 1.2215356731044336, (17, 1) = 2.5862348892169664, (17, 2) = 1.0329309045427477, (18, 1) = 2.7197502962411266, (18, 2) = .8522354096629391, (19, 1) = 2.868097308964122, (19, 2) = .7002166019075773, (20, 1) = 2.9670989492898685, (20, 2) = .6306046195751053, (21, 1) = 3.0272505993270076, (21, 2) = .6023260373483883, (22, 1) = 3.065868563957721, (22, 2) = .5901748294236376, (23, 1) = 3.095251805141905, (23, 2) = .5841893523046803, (24, 1) = 3.11990600006039, (24, 2) = .5813793052108006, (25, 1) = 3.14159265358979, (25, 2) = .5805900104831727}, datatype = float[8], order = C_order); YP := Matrix(25, 2, {(1, 1) = 8.027222261980723, (1, 2) = -.0, (2, 1) = 7.828369059489753, (2, 2) = -10.861805181334308, (3, 1) = 7.287670355233425, (3, 2) = -18.492634875073744, (4, 1) = 6.54481306582631, (4, 2) = -21.694990694475745, (5, 1) = 5.745156668353998, (5, 2) = -21.40568916672294, (6, 1) = 4.987033344028576, (6, 2) = -19.233324103734176, (7, 1) = 4.316682182254274, (7, 2) = -16.42945105476428, (8, 1) = 3.739193866869814, (8, 2) = -13.650236112103359, (9, 1) = 3.241862801797417, (9, 2) = -11.143337685208598, (10, 1) = 2.817246508505084, (10, 2) = -9.007662429827185, (11, 1) = 2.455373052357135, (11, 2) = -7.242681944771625, (12, 1) = 2.146059224600939, (12, 2) = -5.805737354677115, (13, 1) = 1.8786440703412244, (13, 2) = -4.636510741573245, (14, 1) = 1.6409272186274388, (14, 2) = -3.6671563351547207, (15, 1) = 1.4249780044463283, (15, 2) = -2.852577215638926, (16, 1) = 1.2215356731044336, (16, 2) = -2.1483661147204827, (17, 1) = 1.0329309045427477, (17, 2) = -1.5518631600869446, (18, 1) = .8522354096629391, (18, 2) = -1.024669368486231, (19, 1) = .7002166019075773, (19, 2) = -.5874907987071298, (20, 1) = .6306046195751053, (20, 2) = -.35417872498610686, (21, 1) = .6023260373483883, (21, 2) = -.22683550784375403, (22, 1) = .5901748294236376, (22, 2) = -.14876072185358538, (23, 1) = .5841893523046803, (23, 2) = -0.9060027226914852e-1, (24, 1) = .5813793052108006, (24, 2) = -0.4230373846316707e-1, (25, 1) = .5805900104831727, (25, 2) = -0.6298834938184995e-14}, datatype = float[8], order = C_order); errproc := proc (x_bvp) local outpoint, X, Y, yout, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; Digits := 15; outpoint := evalf(x_bvp); X := Vector(25, {(1) = .0, (2) = 0.3556697983665059e-1, (3) = 0.7150815650209673e-1, (4) = .1078355952208751, (5) = .14456202386531472, (6) = .18170088505495013, (7) = .2192663936438809, (8) = .25772582968519775, (9) = .2979698369946872, (10) = .34028667559059855, (11) = .38503401227598166, (12) = .43268598844062967, (13) = .48418267682069066, (14) = .54178398151824, (15) = .6084908129577877, (16) = .6905705623174822, (17) = .7936582413926776, (18) = .9362490342852545, (19) = 1.1290955386275305, (20) = 1.2785289405368818, (21) = 1.3762683057594027, (22) = 1.4410821559755675, (23) = 1.4911439441371346, (24) = 1.5334604576293163, (25) = 1.5707963267949}, datatype = float[8], order = C_order); Y := Matrix(25, 2, {(1, 1) = .0, (1, 2) = 0.31500831778913094e-6, (2, 1) = 0.3132220020535465e-7, (2, 2) = 0.7779131439750735e-7, (3, 1) = 0.14325613236197187e-7, (3, 2) = -0.6898540876289522e-7, (4, 1) = -0.6036355453398498e-8, (4, 2) = 0.7943001698758059e-7, (5, 1) = -0.13260007889026392e-8, (5, 2) = 0.11756415179265621e-6, (6, 1) = 0.1071471554168767e-7, (6, 2) = 0.44460301809323215e-7, (7, 1) = 0.16948724685756332e-7, (7, 2) = -0.6994714895790386e-8, (8, 1) = 0.1782055631368677e-7, (8, 2) = -0.1641681279404671e-7, (9, 1) = 0.16393950975769278e-7, (9, 2) = -0.58427989726364065e-8, (10, 1) = 0.14618894539761003e-7, (10, 2) = 0.7611526601257115e-8, (11, 1) = 0.13363200796132462e-7, (11, 2) = 0.16937803808645205e-7, (12, 1) = 0.12801606379592921e-7, (12, 2) = 0.21469959634400602e-7, (13, 1) = 0.12794426509551367e-7, (13, 2) = 0.22808086897112244e-7, (14, 1) = 0.13080732166085607e-7, (14, 2) = 0.22688679611190916e-7, (15, 1) = 0.13483594578904676e-7, (15, 2) = 0.2185857499871233e-7, (16, 1) = 0.13492144691744485e-7, (16, 2) = 0.2072575098426527e-7, (17, 1) = 0.12840479015099201e-7, (17, 2) = 0.17970028302554435e-7, (18, 1) = 0.720289485656478e-8, (18, 2) = 0.11283105136137962e-7, (19, 1) = -0.10626354523447358e-8, (19, 2) = -0.7464659107819674e-8, (20, 1) = 0.19375865427224214e-8, (20, 2) = -0.1047570220567349e-7, (21, 1) = 0.15016966694503165e-8, (21, 2) = -0.914149161086985e-8, (22, 1) = 0.10081859027200914e-8, (22, 2) = -0.848749960193092e-8, (23, 1) = 0.618147262744349e-9, (23, 2) = -0.8172877515951791e-8, (24, 1) = 0.2897545785236149e-9, (24, 2) = -0.8027811845254448e-8, (25, 1) = .0, (25, 2) = -0.7987456326182811e-8}, datatype = float[8], order = C_order); if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "right" then return X[25] elif outpoint = "order" then return 6 elif outpoint = "error" then return HFloat(3.1500831778913094e-7) elif outpoint = "errorproc" then error "this is already the error procedure" elif outpoint = "rawdata" then return [2, 25, [v(u), diff(v(u), u)], X, Y] else return ('procname')(x_bvp) end if end if; if outpoint < X[1] or X[25] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[25] end if; V := array([1 = 4, 2 = 0]); if Digits <= trunc(evalhf(Digits)) then L := Vector(4, 'datatype' = 'float'[8]); yout := Vector(2, 'datatype' = 'float'[8]); evalhf(`dsolve/numeric/lagrange`(25, 2, X, Y, outpoint, var(yout), var(L), var(V))) else L := Vector(4, 'datatype' = 'sfloat'); yout := Vector(2, 'datatype' = 'sfloat'); `dsolve/numeric/lagrange`(25, 2, X, Y, outpoint, yout, L, V) end if; [u = outpoint, seq('[v(u), diff(v(u), u)]'[i] = yout[i], i = 1 .. 2)] end proc; if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "method" then return "bvp" elif outpoint = "right" then return X[25] elif outpoint = "order" then return 6 elif outpoint = "error" then return HFloat(3.1500831778913094e-7) elif outpoint = "errorproc" then return eval(errproc) elif outpoint = "rawdata" then return [2, 25, "depnames", X, Y, YP] else error "non-numeric value" end if end if; if outpoint < X[1] or X[25] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[25] end if; if Digits <= trunc(evalhf(Digits)) and (_EnvInFsolve <> true or _EnvDSNumericSaveDigits <= trunc(evalhf(Digits))) then V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false) ] ); L := Matrix(7, 2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0, (3, 1) = .0, (3, 2) = .0, (4, 1) = .0, (4, 2) = .0, (5, 1) = .0, (5, 2) = .0, (6, 1) = .0, (6, 2) = .0, (7, 1) = .0, (7, 2) = .0}, datatype = float[8], order = C_order); yout := Vector(2, {(1) = .0, (2) = .0}, datatype = float[8]); evalhf(`dsolve/numeric/hermite`(25, 2, X, Y, YP, outpoint, var(yout), var(L), var(V))) else if _EnvInFsolve = true then Digits := _EnvDSNumericSaveDigits end if; V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false) ] ); L := Matrix(7, 2, {(1, 1) = 0., (1, 2) = 0., (2, 1) = 0., (2, 2) = 0., (3, 1) = 0., (3, 2) = 0., (4, 1) = 0., (4, 2) = 0., (5, 1) = 0., (5, 2) = 0., (6, 1) = 0., (6, 2) = 0., (7, 1) = 0., (7, 2) = 0.}, order = C_order); yout := Vector(2, {(1) = 0., (2) = 0.}); `dsolve/numeric/hermite`(25, 2, X, Y, YP, outpoint, yout, L, V) end if; [outpoint, seq(yout[i], i = 1 .. 2)] end proc, (2) = Array(1..3, {(1) = 36893628625364672284, (2) = 36893628625364664316, (3) = 36893628625364664492}), (3) = [u, v(u), diff(v(u), u)], (4) = 0}); solnproc := data[1]; if not type(outpoint, 'numeric') then if outpoint = "solnprocedure" then return eval(solnproc) elif member(outpoint, ["start", "left", "right", "errorproc", "rawdata", "order", "error"]) then return solnproc(u) elif outpoint = "sysvars" then return data[3] elif procname <> unknown then return ('procname')(u) else `D(v)` := pointto(data[2][3]); return ('`D(v)`')(u) end if end if; try res := solnproc(outpoint); res[3] catch: error end try end proc]

> 

display(
        torus,
        tubeplot([seq(eval(x(u, v(u)), dsol2))], u=0..Pi/2, color=red, radius=0.1),
        pointplot3d(eval([x(0,v(0)),x(Pi/2,v(Pi/2))], dsol2), symbol=solidsphere, symbolsize=20, color="Orange"),
lightmodel=light4, thickness=5, orientation=[-65,65,0], size=[700,400],
style=surface, axes=none);

> 

 

> 

 

Example 3

 

The end points in this example are identical to those of Example 1 but the new geodesic,
drawn in cyan, is different.  In fact, there are infinitely many  geodesics that connect

those two points.

> 

bc3 := v(0)=0,  v(Pi/2)=5*Pi/2;

v(0) = 0, v((1/2)*Pi) = (5/2)*Pi

> 

dsol3 := dsolve({de, bc3}, numeric, output=operator);

[u = proc (u) local _res, _dat, _solnproc; option `Copyright (c) 1993 by the University of Waterloo. All rights reserved.`; _dat := Array(1..4, {(1) = proc (outpoint) local X, Y, YP, yout, errproc, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; X := Vector(59, {(1) = .0, (2) = 0.13680812436809657e-1, (3) = 0.27971051145644167e-1, (4) = 0.4296023135974479e-1, (5) = 0.5851786182852681e-1, (6) = 0.7435973609701907e-1, (7) = 0.9049963296106847e-1, (8) = .1070194360291177, (9) = .12400404122395811, (10) = .14149911102616744, (11) = .15962080170244286, (12) = .17846661163773977, (13) = .1981659899247577, (14) = .2189484735830971, (15) = .2410342106109193, (16) = .2648591917099649, (17) = .29096256417430655, (18) = .3201405168424954, (19) = .3537506263213988, (20) = .39359925436482457, (21) = .44328213017143814, (22) = .508829664760916, (23) = .5993459008179163, (24) = .7156934169226271, (25) = .8282966673508598, (26) = .9130409367878883, (27) = .9748398788122531, (28) = 1.0223008015052495, (29) = 1.0606661846616776, (30) = 1.0932188074676115, (31) = 1.1216707736458404, (32) = 1.1472399927159063, (33) = 1.1706725764522086, (34) = 1.1924372702961963, (35) = 1.2129338175389266, (36) = 1.2323858241405192, (37) = 1.250955152949817, (38) = 1.268795762648431, (39) = 1.2859897085262963, (40) = 1.3026146566339305, (41) = 1.3187452976919192, (42) = 1.3344237627757964, (43) = 1.3497030677053556, (44) = 1.3646567912885574, (45) = 1.3793073520205892, (46) = 1.3936816310415334, (47) = 1.407863413800478, (48) = 1.421874455055364, (49) = 1.4357209480594326, (50) = 1.449452395680341, (51) = 1.463105461314161, (52) = 1.4766814720936998, (53) = 1.4901926102416252, (54) = 1.5036746366401115, (55) = 1.5171303716739315, (56) = 1.5305608572874234, (57) = 1.5439797161303026, (58) = 1.5573915356105725, (59) = 1.5707963267949}, datatype = float[8], order = C_order); Y := Matrix(59, 2, {(1, 1) = .0, (1, 2) = 12.679746440910824, (2, 1) = .17295641157725689, (2, 2) = 12.567764844010108, (3, 1) = .3503687425101184, (3, 2) = 12.226735465168346, (4, 1) = .5296761539769358, (4, 2) = 11.66918438963109, (5, 1) = .7057258717088374, (5, 2) = 10.944263938053027, (6, 1) = .8727045160581467, (6, 2) = 10.12940781981862, (7, 1) = 1.0293675300859457, (7, 2) = 9.286027153268176, (8, 1) = 1.1758543805831463, (8, 2) = 8.457430747175952, (9, 1) = 1.3127197264585053, (9, 2) = 7.671647779381384, (10, 1) = 1.4404597517069202, (10, 2) = 6.945889507639623, (11, 1) = 1.5602083118771684, (11, 2) = 6.28529164237151, (12, 1) = 1.6729138904127339, (12, 2) = 5.690377357385402, (13, 1) = 1.7796245072672423, (13, 2) = 5.157666928630136, (14, 1) = 1.8817159269340347, (14, 2) = 4.680400525330798, (15, 1) = 1.9802286821314778, (15, 2) = 4.2529853303192855, (16, 1) = 2.0768238867507085, (16, 2) = 3.8675310705673276, (17, 1) = 2.173068367774721, (17, 2) = 3.5178917521371798, (18, 1) = 2.270890048473742, (18, 2) = 3.1984139900960797, (19, 1) = 2.3732345057338295, (19, 2) = 2.9030738541180425, (20, 1) = 2.4832304722226684, (20, 2) = 2.6296687269685846, (21, 1) = 2.607217350093064, (21, 2) = 2.3751465147467514, (22, 1) = 2.7547856154152726, (22, 2) = 2.144274550259719, (23, 1) = 2.939616093527199, (23, 2) = 1.961813066469559, (24, 1) = 3.162337834655888, (24, 2) = 1.8947028342219836, (25, 1) = 3.3794261487929442, (25, 2) = 1.9881175251450118, (26, 1) = 3.5551825581028798, (26, 2) = 2.1804710450280265, (27, 1) = 3.69665130286631, (27, 2) = 2.41371269610685, (28, 1) = 3.8169541259171704, (28, 2) = 2.6689707564501304, (29, 1) = 3.9243694765971413, (29, 2) = 2.9422812417392152, (30, 1) = 4.024767234931778, (30, 2) = 3.237123523436467, (31, 1) = 4.121259324516239, (31, 2) = 3.5565852574603505, (32, 1) = 4.216529428559537, (32, 2) = 3.9064929391469123, (33, 1) = 4.312457356149003, (33, 2) = 4.292723456572624, (34, 1) = 4.410412203730131, (34, 2) = 4.720850296041311, (35, 1) = 4.511931211523759, (35, 2) = 5.198255733142234, (36, 1) = 4.618090131180544, (36, 2) = 5.730648052338743, (37, 1) = 4.729865319802914, (37, 2) = 6.322579877258892, (38, 1) = 4.848375884647389, (38, 2) = 6.977624814470957, (39, 1) = 4.974387707359408, (39, 2) = 7.694131606716463, (40, 1) = 5.108607237299307, (40, 2) = 8.464739698798486, (41, 1) = 5.251606222026279, (41, 2) = 9.273734329435953, (42, 1) = 5.403405586640311, (42, 2) = 10.092710388426758, (43, 1) = 5.563684034528413, (43, 2) = 10.88121631033313, (44, 1) = 5.731806698053641, (44, 2) = 11.588598491150355, (45, 1) = 5.905935538945928, (45, 2) = 12.156134835319502, (46, 1) = 6.0836133491883215, (46, 2) = 12.530876289186718, (47, 1) = 6.262647078633722, (47, 2) = 12.678160168039451, (48, 1) = 6.439921199996006, (48, 2) = 12.587708080348033, (49, 1) = 6.612315033049238, (49, 2) = 12.2791039631489, (50, 1) = 6.777770156025198, (50, 2) = 11.79378728978802, (51, 1) = 6.934742040084409, (51, 2) = 11.183836634124887, (52, 1) = 7.0820078226929555, (52, 2) = 10.502833198232775, (53, 1) = 7.219149840143932, (53, 2) = 9.796155822525524, (54, 1) = 7.34648415514996, (54, 2) = 9.096522907410908, (55, 1) = 7.464338781940853, (55, 2) = 8.427075143053173, (56, 1) = 7.573265028033481, (56, 2) = 7.801521511288004, (57, 1) = 7.67403328840954, (57, 2) = 7.22588851589702, (58, 1) = 7.767374215067717, (58, 2) = 6.701837954309735, (59, 1) = 7.85398163397448, (59, 2) = 6.228168238234758}, datatype = float[8], order = C_order); YP := Matrix(59, 2, {(1, 1) = 12.679746440910824, (1, 2) = -.0, (2, 1) = 12.567764844010108, (2, 2) = -16.198974726732907, (3, 1) = 12.226735465168346, (3, 2) = -31.019959257656904, (4, 1) = 11.66918438963109, (4, 2) = -42.61556998298658, (5, 1) = 10.944263938053027, (5, 2) = -49.758388561480395, (6, 1) = 10.12940781981862, (6, 2) = -52.428967468700804, (7, 1) = 9.286027153268176, (7, 2) = -51.59782241363348, (8, 1) = 8.457430747175952, (8, 2) = -48.43560278055239, (9, 1) = 7.671647779381384, (9, 2) = -43.97106435599, (10, 1) = 6.945889507639623, (10, 2) = -38.985423814257885, (11, 1) = 6.28529164237151, (11, 2) = -33.97931379570107, (12, 1) = 5.690377357385402, (12, 2) = -29.25037640167964, (13, 1) = 5.157666928630136, (13, 2) = -24.944278873434246, (14, 1) = 4.680400525330798, (14, 2) = -21.099971062135964, (15, 1) = 4.2529853303192855, (15, 2) = -17.71654087434397, (16, 1) = 3.8675310705673276, (16, 2) = -14.74668416472017, (17, 1) = 3.5178917521371798, (17, 2) = -12.14221532932914, (18, 1) = 3.1984139900960797, (18, 2) = -9.851136892783515, (19, 1) = 2.9030738541180425, (19, 2) = -7.815047340827251, (20, 1) = 2.6296687269685846, (20, 2) = -5.996847641998252, (21, 1) = 2.3751465147467514, (21, 2) = -4.340048329343317, (22, 1) = 2.144274550259719, (22, 2) = -2.7978073376293024, (23, 1) = 1.961813066469559, (23, 2) = -1.320655805545758, (24, 1) = 1.8947028342219836, (24, 2) = .13039610824419176, (25, 1) = 1.9881175251450118, (25, 2) = 1.5787914768274436, (26, 1) = 2.1804710450280265, (26, 2) = 3.0488939854293733, (27, 1) = 2.41371269610685, (27, 2) = 4.590848716067672, (28, 1) = 2.6689707564501304, (28, 2) = 6.254836311594842, (29, 1) = 2.9422812417392152, (29, 2) = 8.080806575409401, (30, 1) = 3.237123523436467, (30, 2) = 10.12389716847413, (31, 1) = 3.5565852574603505, (31, 2) = 12.425685648274625, (32, 1) = 3.9064929391469123, (32, 2) = 15.042554769209124, (33, 1) = 4.292723456572624, (33, 2) = 18.027628684062055, (34, 1) = 4.720850296041311, (34, 2) = 21.42377820734578, (35, 1) = 5.198255733142234, (35, 2) = 25.27273224998513, (36, 1) = 5.730648052338743, (36, 2) = 29.574358687095494, (37, 1) = 6.322579877258892, (37, 2) = 34.27026010854073, (38, 1) = 6.977624814470957, (38, 2) = 39.21645062769387, (39, 1) = 7.694131606716463, (39, 2) = 44.11406880450838, (40, 1) = 8.464739698798486, (40, 2) = 48.471285191096335, (41, 1) = 9.273734329435953, (41, 2) = 51.56564318350719, (42, 1) = 10.092710388426758, (42, 2) = 52.45733113498812, (43, 1) = 10.88121631033313, (43, 2) = 50.12848157993341, (44, 1) = 11.588598491150355, (44, 2) = 43.73470437900077, (45, 1) = 12.156134835319502, (45, 2) = 33.00620482181849, (46, 1) = 12.530876289186718, (46, 2) = 18.57982197925053, (47, 1) = 12.678160168039451, (47, 2) = 1.9582711648426194, (48, 1) = 12.587708080348033, (48, 2) = -14.727409356953261, (49, 1) = 12.2791039631489, (49, 2) = -29.39558090446206, (50, 1) = 11.79378728978802, (50, 2) = -40.669372493045, (51, 1) = 11.183836634124887, (51, 2) = -48.03607321421275, (52, 1) = 10.502833198232775, (52, 2) = -51.72939943874306, (53, 1) = 9.796155822525524, (53, 2) = -52.452900134134595, (54, 1) = 9.096522907410908, (54, 2) = -51.04757024201683, (55, 1) = 8.427075143053173, (55, 2) = -48.28611154332724, (56, 1) = 7.801521511288004, (56, 2) = -44.78604220693657, (57, 1) = 7.22588851589702, (57, 2) = -40.986432776247455, (58, 1) = 6.701837954309735, (58, 2) = -37.176690679157566, (59, 1) = 6.228168238234758, (59, 2) = -33.53206368300497}, datatype = float[8], order = C_order); errproc := proc (x_bvp) local outpoint, X, Y, yout, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; Digits := 15; outpoint := evalf(x_bvp); X := Vector(59, {(1) = .0, (2) = 0.13680812436809657e-1, (3) = 0.27971051145644167e-1, (4) = 0.4296023135974479e-1, (5) = 0.5851786182852681e-1, (6) = 0.7435973609701907e-1, (7) = 0.9049963296106847e-1, (8) = .1070194360291177, (9) = .12400404122395811, (10) = .14149911102616744, (11) = .15962080170244286, (12) = .17846661163773977, (13) = .1981659899247577, (14) = .2189484735830971, (15) = .2410342106109193, (16) = .2648591917099649, (17) = .29096256417430655, (18) = .3201405168424954, (19) = .3537506263213988, (20) = .39359925436482457, (21) = .44328213017143814, (22) = .508829664760916, (23) = .5993459008179163, (24) = .7156934169226271, (25) = .8282966673508598, (26) = .9130409367878883, (27) = .9748398788122531, (28) = 1.0223008015052495, (29) = 1.0606661846616776, (30) = 1.0932188074676115, (31) = 1.1216707736458404, (32) = 1.1472399927159063, (33) = 1.1706725764522086, (34) = 1.1924372702961963, (35) = 1.2129338175389266, (36) = 1.2323858241405192, (37) = 1.250955152949817, (38) = 1.268795762648431, (39) = 1.2859897085262963, (40) = 1.3026146566339305, (41) = 1.3187452976919192, (42) = 1.3344237627757964, (43) = 1.3497030677053556, (44) = 1.3646567912885574, (45) = 1.3793073520205892, (46) = 1.3936816310415334, (47) = 1.407863413800478, (48) = 1.421874455055364, (49) = 1.4357209480594326, (50) = 1.449452395680341, (51) = 1.463105461314161, (52) = 1.4766814720936998, (53) = 1.4901926102416252, (54) = 1.5036746366401115, (55) = 1.5171303716739315, (56) = 1.5305608572874234, (57) = 1.5439797161303026, (58) = 1.5573915356105725, (59) = 1.5707963267949}, datatype = float[8], order = C_order); Y := Matrix(59, 2, {(1, 1) = .0, (1, 2) = -0.2040971830865995e-12, (2, 1) = -0.2515085127931946e-15, (2, 2) = -0.2182240868305703e-12, (3, 1) = -0.3034400215680231e-14, (3, 2) = -0.24629572182126297e-12, (4, 1) = -0.70293294530316345e-14, (4, 2) = -0.27355561891671303e-12, (5, 1) = -0.11642509457266535e-13, (5, 2) = -0.2921725988376451e-12, (6, 1) = -0.14891698174826332e-13, (6, 2) = -0.27332966679506926e-12, (7, 1) = -0.2244158402930937e-13, (7, 2) = -0.21555448086863683e-12, (8, 1) = -0.23830070405255755e-13, (8, 2) = -0.14452163468605652e-12, (9, 1) = -0.28784241689197694e-13, (9, 2) = -0.8936429016769738e-13, (10, 1) = -0.2870763992607896e-13, (10, 2) = -0.7015153701410498e-13, (11, 1) = -0.29805232120338505e-13, (11, 2) = -0.20455226198960395e-13, (12, 1) = -0.3071869736898188e-13, (12, 2) = -0.949608712395301e-14, (13, 1) = -0.33085750279284234e-13, (13, 2) = -0.9853725522439467e-14, (14, 1) = -0.3024025650769599e-13, (14, 2) = 0.55343272382320285e-14, (15, 1) = -0.2989762453456761e-13, (15, 2) = 0.9484091577415516e-14, (16, 1) = -0.3126420836975352e-13, (16, 2) = 0.10993914919372806e-15, (17, 1) = -0.2904952409517121e-13, (17, 2) = 0.7741875869166384e-14, (18, 1) = -0.26389405390354672e-13, (18, 2) = -0.39736138956257974e-14, (19, 1) = -0.3066642976676729e-13, (19, 2) = -0.8841791351636723e-14, (20, 1) = -0.3381874764737987e-13, (20, 2) = -0.10474029579197825e-13, (21, 1) = -0.3076950762715862e-13, (21, 2) = -0.14630357321113296e-13, (22, 1) = -0.25742990878046857e-13, (22, 2) = -0.32013707096004785e-13, (23, 1) = -0.26034531106847924e-14, (23, 2) = -0.17882398235769704e-13, (24, 1) = -0.24294629294277565e-13, (24, 2) = 0.6033751098446911e-13, (25, 1) = -0.10212795741481291e-13, (25, 2) = -0.5409909790245115e-14, (26, 1) = 0.65234775286404595e-14, (26, 2) = 0.10506976038853487e-14, (27, 1) = 0.15145921627767115e-13, (27, 2) = 0.11062582034309036e-13, (28, 1) = 0.17727034515154246e-13, (28, 2) = 0.1392374003706416e-13, (29, 1) = 0.20190304444405087e-13, (29, 2) = 0.24077662756883795e-13, (30, 1) = 0.15567756982704383e-13, (30, 2) = 0.3220356728033926e-13, (31, 1) = 0.2049236289955625e-13, (31, 2) = 0.4582315826902611e-13, (32, 1) = 0.2093779309445625e-13, (32, 2) = 0.50847503530540665e-13, (33, 1) = 0.34656722670579105e-13, (33, 2) = 0.65791408284367e-13, (34, 1) = 0.1414627403891834e-13, (34, 2) = 0.7434970442839986e-13, (35, 1) = 0.1430197385087178e-13, (35, 2) = 0.807995248637132e-13, (36, 1) = 0.2674246752632126e-13, (36, 2) = 0.8238316242997303e-13, (37, 1) = 0.3749521089102351e-13, (37, 2) = 0.9551972331079688e-13, (38, 1) = 0.43798173157104124e-13, (38, 2) = 0.8882243056529444e-13, (39, 1) = 0.34765247192333954e-13, (39, 2) = 0.7174268116999525e-13, (40, 1) = 0.3492718539460031e-13, (40, 2) = 0.25074509668472094e-13, (41, 1) = 0.3268343950774218e-13, (41, 2) = -0.6182099171300564e-13, (42, 1) = 0.7615127467050659e-14, (42, 2) = -0.1854246486047524e-12, (43, 1) = 0.20280258788197414e-13, (43, 2) = -0.2612655964015281e-12, (44, 1) = 0.5268977580072408e-14, (44, 2) = -0.3058713868413555e-12, (45, 1) = 0.8243135377351008e-14, (45, 2) = -0.3000324869932838e-12, (46, 1) = 0.9651203882669847e-14, (46, 2) = -0.23958986839330786e-12, (47, 1) = -0.15172603499122148e-13, (47, 2) = -0.14114952424844551e-12, (48, 1) = 0.6415293117671377e-14, (48, 2) = -0.3574626525598719e-13, (49, 1) = 0.11771473310921094e-13, (49, 2) = 0.12758808412240534e-13, (50, 1) = 0.350361321759921e-14, (50, 2) = 0.18960251337597162e-13, (51, 1) = -0.402567915071639e-14, (51, 2) = -0.42598728779319296e-13, (52, 1) = 0.14493706543893737e-13, (52, 2) = -0.4380811741884113e-13, (53, 1) = 0.19531436506149837e-13, (53, 2) = -0.8490114617635297e-13, (54, 1) = 0.4784180312117899e-14, (54, 2) = -0.10238102255674353e-12, (55, 1) = 0.13186433513432542e-13, (55, 2) = -0.12142340951879697e-12, (56, 1) = 0.68589188913736246e-14, (56, 2) = -0.10734720880680753e-12, (57, 1) = 0.4428481028761533e-14, (57, 2) = -0.9866461770792047e-13, (58, 1) = 0.7387414481358141e-14, (58, 2) = -0.9166293971010342e-13, (59, 1) = .0, (59, 2) = -0.731647525953621e-13}, datatype = float[8], order = C_order); if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "right" then return X[59] elif outpoint = "order" then return 10 elif outpoint = "error" then return HFloat(3.058713868413555e-13) elif outpoint = "errorproc" then error "this is already the error procedure" elif outpoint = "rawdata" then return [2, 59, [v(u), diff(v(u), u)], X, Y] else return ('procname')(x_bvp) end if end if; if outpoint < X[1] or X[59] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[59] end if; V := array([1 = 4, 2 = 0]); if Digits <= trunc(evalhf(Digits)) then L := Vector(4, 'datatype' = 'float'[8]); yout := Vector(2, 'datatype' = 'float'[8]); evalhf(`dsolve/numeric/lagrange`(59, 2, X, Y, outpoint, var(yout), var(L), var(V))) else L := Vector(4, 'datatype' = 'sfloat'); yout := Vector(2, 'datatype' = 'sfloat'); `dsolve/numeric/lagrange`(59, 2, X, Y, outpoint, yout, L, V) end if; [u = outpoint, seq('[v(u), diff(v(u), u)]'[i] = yout[i], i = 1 .. 2)] end proc; if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "method" then return "bvp" elif outpoint = "right" then return X[59] elif outpoint = "order" then return 10 elif outpoint = "error" then return HFloat(3.058713868413555e-13) elif outpoint = "errorproc" then return eval(errproc) elif outpoint = "rawdata" then return [2, 59, "depnames", X, Y, YP] else error "non-numeric value" end if end if; if outpoint < X[1] or X[59] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[59] end if; if Digits <= trunc(evalhf(Digits)) and (_EnvInFsolve <> true or _EnvDSNumericSaveDigits <= trunc(evalhf(Digits))) then V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false) ] ); L := Matrix(7, 2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0, (3, 1) = .0, (3, 2) = .0, (4, 1) = .0, (4, 2) = .0, (5, 1) = .0, (5, 2) = .0, (6, 1) = .0, (6, 2) = .0, (7, 1) = .0, (7, 2) = .0}, datatype = float[8], order = C_order); yout := Vector(2, {(1) = .0, (2) = .0}, datatype = float[8]); evalhf(`dsolve/numeric/hermite`(59, 2, X, Y, YP, outpoint, var(yout), var(L), var(V))) else if _EnvInFsolve = true then Digits := _EnvDSNumericSaveDigits end if; V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false) ] ); L := Matrix(7, 2, {(1, 1) = 0., (1, 2) = 0., (2, 1) = 0., (2, 2) = 0., (3, 1) = 0., (3, 2) = 0., (4, 1) = 0., (4, 2) = 0., (5, 1) = 0., (5, 2) = 0., (6, 1) = 0., (6, 2) = 0., (7, 1) = 0., (7, 2) = 0.}, order = C_order); yout := Vector(2, {(1) = 0., (2) = 0.}); `dsolve/numeric/hermite`(59, 2, X, Y, YP, outpoint, yout, L, V) end if; [outpoint, seq(yout[i], i = 1 .. 2)] end proc, (2) = Array(1..3, {(1) = 36893628625353841228, (2) = 36893628625353841404, (3) = 36893628625353841580}), (3) = [u, v(u), diff(v(u), u)], (4) = 0}); _solnproc := _dat[1]; if member(u, ["last", 'last']) then _res := _solnproc("last"); if type(_res, 'list') then return _res[1] end if elif type(u, `=`) and member(lhs(u), ["initial", 'initial']) then if type(rhs(u), 'list') then _res := _solnproc("initial" = [0, op(rhs(u))]) else _res := _solnproc("initial" = [1, rhs(u)]) end if; if type(_res, 'list') then return _res[1] end if elif u = "sysvars" then return _dat[3] end if; u end proc, v = proc (u) local res, data, solnproc, v, outpoint; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then outpoint := evalf[_EnvDSNumericSaveDigits](u) else outpoint := evalf(u) end if; data := Array(1..4, {(1) = proc (outpoint) local X, Y, YP, yout, errproc, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; X := Vector(59, {(1) = .0, (2) = 0.13680812436809657e-1, (3) = 0.27971051145644167e-1, (4) = 0.4296023135974479e-1, (5) = 0.5851786182852681e-1, (6) = 0.7435973609701907e-1, (7) = 0.9049963296106847e-1, (8) = .1070194360291177, (9) = .12400404122395811, (10) = .14149911102616744, (11) = .15962080170244286, (12) = .17846661163773977, (13) = .1981659899247577, (14) = .2189484735830971, (15) = .2410342106109193, (16) = .2648591917099649, (17) = .29096256417430655, (18) = .3201405168424954, (19) = .3537506263213988, (20) = .39359925436482457, (21) = .44328213017143814, (22) = .508829664760916, (23) = .5993459008179163, (24) = .7156934169226271, (25) = .8282966673508598, (26) = .9130409367878883, (27) = .9748398788122531, (28) = 1.0223008015052495, (29) = 1.0606661846616776, (30) = 1.0932188074676115, (31) = 1.1216707736458404, (32) = 1.1472399927159063, (33) = 1.1706725764522086, (34) = 1.1924372702961963, (35) = 1.2129338175389266, (36) = 1.2323858241405192, (37) = 1.250955152949817, (38) = 1.268795762648431, (39) = 1.2859897085262963, (40) = 1.3026146566339305, (41) = 1.3187452976919192, (42) = 1.3344237627757964, (43) = 1.3497030677053556, (44) = 1.3646567912885574, (45) = 1.3793073520205892, (46) = 1.3936816310415334, (47) = 1.407863413800478, (48) = 1.421874455055364, (49) = 1.4357209480594326, (50) = 1.449452395680341, (51) = 1.463105461314161, (52) = 1.4766814720936998, (53) = 1.4901926102416252, (54) = 1.5036746366401115, (55) = 1.5171303716739315, (56) = 1.5305608572874234, (57) = 1.5439797161303026, (58) = 1.5573915356105725, (59) = 1.5707963267949}, datatype = float[8], order = C_order); Y := Matrix(59, 2, {(1, 1) = .0, (1, 2) = 12.679746440910824, (2, 1) = .17295641157725689, (2, 2) = 12.567764844010108, (3, 1) = .3503687425101184, (3, 2) = 12.226735465168346, (4, 1) = .5296761539769358, (4, 2) = 11.66918438963109, (5, 1) = .7057258717088374, (5, 2) = 10.944263938053027, (6, 1) = .8727045160581467, (6, 2) = 10.12940781981862, (7, 1) = 1.0293675300859457, (7, 2) = 9.286027153268176, (8, 1) = 1.1758543805831463, (8, 2) = 8.457430747175952, (9, 1) = 1.3127197264585053, (9, 2) = 7.671647779381384, (10, 1) = 1.4404597517069202, (10, 2) = 6.945889507639623, (11, 1) = 1.5602083118771684, (11, 2) = 6.28529164237151, (12, 1) = 1.6729138904127339, (12, 2) = 5.690377357385402, (13, 1) = 1.7796245072672423, (13, 2) = 5.157666928630136, (14, 1) = 1.8817159269340347, (14, 2) = 4.680400525330798, (15, 1) = 1.9802286821314778, (15, 2) = 4.2529853303192855, (16, 1) = 2.0768238867507085, (16, 2) = 3.8675310705673276, (17, 1) = 2.173068367774721, (17, 2) = 3.5178917521371798, (18, 1) = 2.270890048473742, (18, 2) = 3.1984139900960797, (19, 1) = 2.3732345057338295, (19, 2) = 2.9030738541180425, (20, 1) = 2.4832304722226684, (20, 2) = 2.6296687269685846, (21, 1) = 2.607217350093064, (21, 2) = 2.3751465147467514, (22, 1) = 2.7547856154152726, (22, 2) = 2.144274550259719, (23, 1) = 2.939616093527199, (23, 2) = 1.961813066469559, (24, 1) = 3.162337834655888, (24, 2) = 1.8947028342219836, (25, 1) = 3.3794261487929442, (25, 2) = 1.9881175251450118, (26, 1) = 3.5551825581028798, (26, 2) = 2.1804710450280265, (27, 1) = 3.69665130286631, (27, 2) = 2.41371269610685, (28, 1) = 3.8169541259171704, (28, 2) = 2.6689707564501304, (29, 1) = 3.9243694765971413, (29, 2) = 2.9422812417392152, (30, 1) = 4.024767234931778, (30, 2) = 3.237123523436467, (31, 1) = 4.121259324516239, (31, 2) = 3.5565852574603505, (32, 1) = 4.216529428559537, (32, 2) = 3.9064929391469123, (33, 1) = 4.312457356149003, (33, 2) = 4.292723456572624, (34, 1) = 4.410412203730131, (34, 2) = 4.720850296041311, (35, 1) = 4.511931211523759, (35, 2) = 5.198255733142234, (36, 1) = 4.618090131180544, (36, 2) = 5.730648052338743, (37, 1) = 4.729865319802914, (37, 2) = 6.322579877258892, (38, 1) = 4.848375884647389, (38, 2) = 6.977624814470957, (39, 1) = 4.974387707359408, (39, 2) = 7.694131606716463, (40, 1) = 5.108607237299307, (40, 2) = 8.464739698798486, (41, 1) = 5.251606222026279, (41, 2) = 9.273734329435953, (42, 1) = 5.403405586640311, (42, 2) = 10.092710388426758, (43, 1) = 5.563684034528413, (43, 2) = 10.88121631033313, (44, 1) = 5.731806698053641, (44, 2) = 11.588598491150355, (45, 1) = 5.905935538945928, (45, 2) = 12.156134835319502, (46, 1) = 6.0836133491883215, (46, 2) = 12.530876289186718, (47, 1) = 6.262647078633722, (47, 2) = 12.678160168039451, (48, 1) = 6.439921199996006, (48, 2) = 12.587708080348033, (49, 1) = 6.612315033049238, (49, 2) = 12.2791039631489, (50, 1) = 6.777770156025198, (50, 2) = 11.79378728978802, (51, 1) = 6.934742040084409, (51, 2) = 11.183836634124887, (52, 1) = 7.0820078226929555, (52, 2) = 10.502833198232775, (53, 1) = 7.219149840143932, (53, 2) = 9.796155822525524, (54, 1) = 7.34648415514996, (54, 2) = 9.096522907410908, (55, 1) = 7.464338781940853, (55, 2) = 8.427075143053173, (56, 1) = 7.573265028033481, (56, 2) = 7.801521511288004, (57, 1) = 7.67403328840954, (57, 2) = 7.22588851589702, (58, 1) = 7.767374215067717, (58, 2) = 6.701837954309735, (59, 1) = 7.85398163397448, (59, 2) = 6.228168238234758}, datatype = float[8], order = C_order); YP := Matrix(59, 2, {(1, 1) = 12.679746440910824, (1, 2) = -.0, (2, 1) = 12.567764844010108, (2, 2) = -16.198974726732907, (3, 1) = 12.226735465168346, (3, 2) = -31.019959257656904, (4, 1) = 11.66918438963109, (4, 2) = -42.61556998298658, (5, 1) = 10.944263938053027, (5, 2) = -49.758388561480395, (6, 1) = 10.12940781981862, (6, 2) = -52.428967468700804, (7, 1) = 9.286027153268176, (7, 2) = -51.59782241363348, (8, 1) = 8.457430747175952, (8, 2) = -48.43560278055239, (9, 1) = 7.671647779381384, (9, 2) = -43.97106435599, (10, 1) = 6.945889507639623, (10, 2) = -38.985423814257885, (11, 1) = 6.28529164237151, (11, 2) = -33.97931379570107, (12, 1) = 5.690377357385402, (12, 2) = -29.25037640167964, (13, 1) = 5.157666928630136, (13, 2) = -24.944278873434246, (14, 1) = 4.680400525330798, (14, 2) = -21.099971062135964, (15, 1) = 4.2529853303192855, (15, 2) = -17.71654087434397, (16, 1) = 3.8675310705673276, (16, 2) = -14.74668416472017, (17, 1) = 3.5178917521371798, (17, 2) = -12.14221532932914, (18, 1) = 3.1984139900960797, (18, 2) = -9.851136892783515, (19, 1) = 2.9030738541180425, (19, 2) = -7.815047340827251, (20, 1) = 2.6296687269685846, (20, 2) = -5.996847641998252, (21, 1) = 2.3751465147467514, (21, 2) = -4.340048329343317, (22, 1) = 2.144274550259719, (22, 2) = -2.7978073376293024, (23, 1) = 1.961813066469559, (23, 2) = -1.320655805545758, (24, 1) = 1.8947028342219836, (24, 2) = .13039610824419176, (25, 1) = 1.9881175251450118, (25, 2) = 1.5787914768274436, (26, 1) = 2.1804710450280265, (26, 2) = 3.0488939854293733, (27, 1) = 2.41371269610685, (27, 2) = 4.590848716067672, (28, 1) = 2.6689707564501304, (28, 2) = 6.254836311594842, (29, 1) = 2.9422812417392152, (29, 2) = 8.080806575409401, (30, 1) = 3.237123523436467, (30, 2) = 10.12389716847413, (31, 1) = 3.5565852574603505, (31, 2) = 12.425685648274625, (32, 1) = 3.9064929391469123, (32, 2) = 15.042554769209124, (33, 1) = 4.292723456572624, (33, 2) = 18.027628684062055, (34, 1) = 4.720850296041311, (34, 2) = 21.42377820734578, (35, 1) = 5.198255733142234, (35, 2) = 25.27273224998513, (36, 1) = 5.730648052338743, (36, 2) = 29.574358687095494, (37, 1) = 6.322579877258892, (37, 2) = 34.27026010854073, (38, 1) = 6.977624814470957, (38, 2) = 39.21645062769387, (39, 1) = 7.694131606716463, (39, 2) = 44.11406880450838, (40, 1) = 8.464739698798486, (40, 2) = 48.471285191096335, (41, 1) = 9.273734329435953, (41, 2) = 51.56564318350719, (42, 1) = 10.092710388426758, (42, 2) = 52.45733113498812, (43, 1) = 10.88121631033313, (43, 2) = 50.12848157993341, (44, 1) = 11.588598491150355, (44, 2) = 43.73470437900077, (45, 1) = 12.156134835319502, (45, 2) = 33.00620482181849, (46, 1) = 12.530876289186718, (46, 2) = 18.57982197925053, (47, 1) = 12.678160168039451, (47, 2) = 1.9582711648426194, (48, 1) = 12.587708080348033, (48, 2) = -14.727409356953261, (49, 1) = 12.2791039631489, (49, 2) = -29.39558090446206, (50, 1) = 11.79378728978802, (50, 2) = -40.669372493045, (51, 1) = 11.183836634124887, (51, 2) = -48.03607321421275, (52, 1) = 10.502833198232775, (52, 2) = -51.72939943874306, (53, 1) = 9.796155822525524, (53, 2) = -52.452900134134595, (54, 1) = 9.096522907410908, (54, 2) = -51.04757024201683, (55, 1) = 8.427075143053173, (55, 2) = -48.28611154332724, (56, 1) = 7.801521511288004, (56, 2) = -44.78604220693657, (57, 1) = 7.22588851589702, (57, 2) = -40.986432776247455, (58, 1) = 6.701837954309735, (58, 2) = -37.176690679157566, (59, 1) = 6.228168238234758, (59, 2) = -33.53206368300497}, datatype = float[8], order = C_order); errproc := proc (x_bvp) local outpoint, X, Y, yout, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; Digits := 15; outpoint := evalf(x_bvp); X := Vector(59, {(1) = .0, (2) = 0.13680812436809657e-1, (3) = 0.27971051145644167e-1, (4) = 0.4296023135974479e-1, (5) = 0.5851786182852681e-1, (6) = 0.7435973609701907e-1, (7) = 0.9049963296106847e-1, (8) = .1070194360291177, (9) = .12400404122395811, (10) = .14149911102616744, (11) = .15962080170244286, (12) = .17846661163773977, (13) = .1981659899247577, (14) = .2189484735830971, (15) = .2410342106109193, (16) = .2648591917099649, (17) = .29096256417430655, (18) = .3201405168424954, (19) = .3537506263213988, (20) = .39359925436482457, (21) = .44328213017143814, (22) = .508829664760916, (23) = .5993459008179163, (24) = .7156934169226271, (25) = .8282966673508598, (26) = .9130409367878883, (27) = .9748398788122531, (28) = 1.0223008015052495, (29) = 1.0606661846616776, (30) = 1.0932188074676115, (31) = 1.1216707736458404, (32) = 1.1472399927159063, (33) = 1.1706725764522086, (34) = 1.1924372702961963, (35) = 1.2129338175389266, (36) = 1.2323858241405192, (37) = 1.250955152949817, (38) = 1.268795762648431, (39) = 1.2859897085262963, (40) = 1.3026146566339305, (41) = 1.3187452976919192, (42) = 1.3344237627757964, (43) = 1.3497030677053556, (44) = 1.3646567912885574, (45) = 1.3793073520205892, (46) = 1.3936816310415334, (47) = 1.407863413800478, (48) = 1.421874455055364, (49) = 1.4357209480594326, (50) = 1.449452395680341, (51) = 1.463105461314161, (52) = 1.4766814720936998, (53) = 1.4901926102416252, (54) = 1.5036746366401115, (55) = 1.5171303716739315, (56) = 1.5305608572874234, (57) = 1.5439797161303026, (58) = 1.5573915356105725, (59) = 1.5707963267949}, datatype = float[8], order = C_order); Y := Matrix(59, 2, {(1, 1) = .0, (1, 2) = -0.2040971830865995e-12, (2, 1) = -0.2515085127931946e-15, (2, 2) = -0.2182240868305703e-12, (3, 1) = -0.3034400215680231e-14, (3, 2) = -0.24629572182126297e-12, (4, 1) = -0.70293294530316345e-14, (4, 2) = -0.27355561891671303e-12, (5, 1) = -0.11642509457266535e-13, (5, 2) = -0.2921725988376451e-12, (6, 1) = -0.14891698174826332e-13, (6, 2) = -0.27332966679506926e-12, (7, 1) = -0.2244158402930937e-13, (7, 2) = -0.21555448086863683e-12, (8, 1) = -0.23830070405255755e-13, (8, 2) = -0.14452163468605652e-12, (9, 1) = -0.28784241689197694e-13, (9, 2) = -0.8936429016769738e-13, (10, 1) = -0.2870763992607896e-13, (10, 2) = -0.7015153701410498e-13, (11, 1) = -0.29805232120338505e-13, (11, 2) = -0.20455226198960395e-13, (12, 1) = -0.3071869736898188e-13, (12, 2) = -0.949608712395301e-14, (13, 1) = -0.33085750279284234e-13, (13, 2) = -0.9853725522439467e-14, (14, 1) = -0.3024025650769599e-13, (14, 2) = 0.55343272382320285e-14, (15, 1) = -0.2989762453456761e-13, (15, 2) = 0.9484091577415516e-14, (16, 1) = -0.3126420836975352e-13, (16, 2) = 0.10993914919372806e-15, (17, 1) = -0.2904952409517121e-13, (17, 2) = 0.7741875869166384e-14, (18, 1) = -0.26389405390354672e-13, (18, 2) = -0.39736138956257974e-14, (19, 1) = -0.3066642976676729e-13, (19, 2) = -0.8841791351636723e-14, (20, 1) = -0.3381874764737987e-13, (20, 2) = -0.10474029579197825e-13, (21, 1) = -0.3076950762715862e-13, (21, 2) = -0.14630357321113296e-13, (22, 1) = -0.25742990878046857e-13, (22, 2) = -0.32013707096004785e-13, (23, 1) = -0.26034531106847924e-14, (23, 2) = -0.17882398235769704e-13, (24, 1) = -0.24294629294277565e-13, (24, 2) = 0.6033751098446911e-13, (25, 1) = -0.10212795741481291e-13, (25, 2) = -0.5409909790245115e-14, (26, 1) = 0.65234775286404595e-14, (26, 2) = 0.10506976038853487e-14, (27, 1) = 0.15145921627767115e-13, (27, 2) = 0.11062582034309036e-13, (28, 1) = 0.17727034515154246e-13, (28, 2) = 0.1392374003706416e-13, (29, 1) = 0.20190304444405087e-13, (29, 2) = 0.24077662756883795e-13, (30, 1) = 0.15567756982704383e-13, (30, 2) = 0.3220356728033926e-13, (31, 1) = 0.2049236289955625e-13, (31, 2) = 0.4582315826902611e-13, (32, 1) = 0.2093779309445625e-13, (32, 2) = 0.50847503530540665e-13, (33, 1) = 0.34656722670579105e-13, (33, 2) = 0.65791408284367e-13, (34, 1) = 0.1414627403891834e-13, (34, 2) = 0.7434970442839986e-13, (35, 1) = 0.1430197385087178e-13, (35, 2) = 0.807995248637132e-13, (36, 1) = 0.2674246752632126e-13, (36, 2) = 0.8238316242997303e-13, (37, 1) = 0.3749521089102351e-13, (37, 2) = 0.9551972331079688e-13, (38, 1) = 0.43798173157104124e-13, (38, 2) = 0.8882243056529444e-13, (39, 1) = 0.34765247192333954e-13, (39, 2) = 0.7174268116999525e-13, (40, 1) = 0.3492718539460031e-13, (40, 2) = 0.25074509668472094e-13, (41, 1) = 0.3268343950774218e-13, (41, 2) = -0.6182099171300564e-13, (42, 1) = 0.7615127467050659e-14, (42, 2) = -0.1854246486047524e-12, (43, 1) = 0.20280258788197414e-13, (43, 2) = -0.2612655964015281e-12, (44, 1) = 0.5268977580072408e-14, (44, 2) = -0.3058713868413555e-12, (45, 1) = 0.8243135377351008e-14, (45, 2) = -0.3000324869932838e-12, (46, 1) = 0.9651203882669847e-14, (46, 2) = -0.23958986839330786e-12, (47, 1) = -0.15172603499122148e-13, (47, 2) = -0.14114952424844551e-12, (48, 1) = 0.6415293117671377e-14, (48, 2) = -0.3574626525598719e-13, (49, 1) = 0.11771473310921094e-13, (49, 2) = 0.12758808412240534e-13, (50, 1) = 0.350361321759921e-14, (50, 2) = 0.18960251337597162e-13, (51, 1) = -0.402567915071639e-14, (51, 2) = -0.42598728779319296e-13, (52, 1) = 0.14493706543893737e-13, (52, 2) = -0.4380811741884113e-13, (53, 1) = 0.19531436506149837e-13, (53, 2) = -0.8490114617635297e-13, (54, 1) = 0.4784180312117899e-14, (54, 2) = -0.10238102255674353e-12, (55, 1) = 0.13186433513432542e-13, (55, 2) = -0.12142340951879697e-12, (56, 1) = 0.68589188913736246e-14, (56, 2) = -0.10734720880680753e-12, (57, 1) = 0.4428481028761533e-14, (57, 2) = -0.9866461770792047e-13, (58, 1) = 0.7387414481358141e-14, (58, 2) = -0.9166293971010342e-13, (59, 1) = .0, (59, 2) = -0.731647525953621e-13}, datatype = float[8], order = C_order); if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "right" then return X[59] elif outpoint = "order" then return 10 elif outpoint = "error" then return HFloat(3.058713868413555e-13) elif outpoint = "errorproc" then error "this is already the error procedure" elif outpoint = "rawdata" then return [2, 59, [v(u), diff(v(u), u)], X, Y] else return ('procname')(x_bvp) end if end if; if outpoint < X[1] or X[59] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[59] end if; V := array([1 = 4, 2 = 0]); if Digits <= trunc(evalhf(Digits)) then L := Vector(4, 'datatype' = 'float'[8]); yout := Vector(2, 'datatype' = 'float'[8]); evalhf(`dsolve/numeric/lagrange`(59, 2, X, Y, outpoint, var(yout), var(L), var(V))) else L := Vector(4, 'datatype' = 'sfloat'); yout := Vector(2, 'datatype' = 'sfloat'); `dsolve/numeric/lagrange`(59, 2, X, Y, outpoint, yout, L, V) end if; [u = outpoint, seq('[v(u), diff(v(u), u)]'[i] = yout[i], i = 1 .. 2)] end proc; if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "method" then return "bvp" elif outpoint = "right" then return X[59] elif outpoint = "order" then return 10 elif outpoint = "error" then return HFloat(3.058713868413555e-13) elif outpoint = "errorproc" then return eval(errproc) elif outpoint = "rawdata" then return [2, 59, "depnames", X, Y, YP] else error "non-numeric value" end if end if; if outpoint < X[1] or X[59] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[59] end if; if Digits <= trunc(evalhf(Digits)) and (_EnvInFsolve <> true or _EnvDSNumericSaveDigits <= trunc(evalhf(Digits))) then V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false) ] ); L := Matrix(7, 2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0, (3, 1) = .0, (3, 2) = .0, (4, 1) = .0, (4, 2) = .0, (5, 1) = .0, (5, 2) = .0, (6, 1) = .0, (6, 2) = .0, (7, 1) = .0, (7, 2) = .0}, datatype = float[8], order = C_order); yout := Vector(2, {(1) = .0, (2) = .0}, datatype = float[8]); evalhf(`dsolve/numeric/hermite`(59, 2, X, Y, YP, outpoint, var(yout), var(L), var(V))) else if _EnvInFsolve = true then Digits := _EnvDSNumericSaveDigits end if; V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false) ] ); L := Matrix(7, 2, {(1, 1) = 0., (1, 2) = 0., (2, 1) = 0., (2, 2) = 0., (3, 1) = 0., (3, 2) = 0., (4, 1) = 0., (4, 2) = 0., (5, 1) = 0., (5, 2) = 0., (6, 1) = 0., (6, 2) = 0., (7, 1) = 0., (7, 2) = 0.}, order = C_order); yout := Vector(2, {(1) = 0., (2) = 0.}); `dsolve/numeric/hermite`(59, 2, X, Y, YP, outpoint, yout, L, V) end if; [outpoint, seq(yout[i], i = 1 .. 2)] end proc, (2) = Array(1..3, {(1) = 36893628625353841228, (2) = 36893628625353841404, (3) = 36893628625353841580}), (3) = [u, v(u), diff(v(u), u)], (4) = 0}); solnproc := data[1]; if not type(outpoint, 'numeric') then if outpoint = "solnprocedure" then return eval(solnproc) elif member(outpoint, ["start", "left", "right", "errorproc", "rawdata", "order", "error"]) then return solnproc(u) elif outpoint = "sysvars" then return data[3] elif procname <> unknown then return ('procname')(u) else v := pointto(data[2][2]); return ('v')(u) end if end if; try res := solnproc(outpoint); res[2] catch: error end try end proc, D(v) = proc (u) local res, data, solnproc, `D(v)`, outpoint; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then outpoint := evalf[_EnvDSNumericSaveDigits](u) else outpoint := evalf(u) end if; data := Array(1..4, {(1) = proc (outpoint) local X, Y, YP, yout, errproc, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; X := Vector(59, {(1) = .0, (2) = 0.13680812436809657e-1, (3) = 0.27971051145644167e-1, (4) = 0.4296023135974479e-1, (5) = 0.5851786182852681e-1, (6) = 0.7435973609701907e-1, (7) = 0.9049963296106847e-1, (8) = .1070194360291177, (9) = .12400404122395811, (10) = .14149911102616744, (11) = .15962080170244286, (12) = .17846661163773977, (13) = .1981659899247577, (14) = .2189484735830971, (15) = .2410342106109193, (16) = .2648591917099649, (17) = .29096256417430655, (18) = .3201405168424954, (19) = .3537506263213988, (20) = .39359925436482457, (21) = .44328213017143814, (22) = .508829664760916, (23) = .5993459008179163, (24) = .7156934169226271, (25) = .8282966673508598, (26) = .9130409367878883, (27) = .9748398788122531, (28) = 1.0223008015052495, (29) = 1.0606661846616776, (30) = 1.0932188074676115, (31) = 1.1216707736458404, (32) = 1.1472399927159063, (33) = 1.1706725764522086, (34) = 1.1924372702961963, (35) = 1.2129338175389266, (36) = 1.2323858241405192, (37) = 1.250955152949817, (38) = 1.268795762648431, (39) = 1.2859897085262963, (40) = 1.3026146566339305, (41) = 1.3187452976919192, (42) = 1.3344237627757964, (43) = 1.3497030677053556, (44) = 1.3646567912885574, (45) = 1.3793073520205892, (46) = 1.3936816310415334, (47) = 1.407863413800478, (48) = 1.421874455055364, (49) = 1.4357209480594326, (50) = 1.449452395680341, (51) = 1.463105461314161, (52) = 1.4766814720936998, (53) = 1.4901926102416252, (54) = 1.5036746366401115, (55) = 1.5171303716739315, (56) = 1.5305608572874234, (57) = 1.5439797161303026, (58) = 1.5573915356105725, (59) = 1.5707963267949}, datatype = float[8], order = C_order); Y := Matrix(59, 2, {(1, 1) = .0, (1, 2) = 12.679746440910824, (2, 1) = .17295641157725689, (2, 2) = 12.567764844010108, (3, 1) = .3503687425101184, (3, 2) = 12.226735465168346, (4, 1) = .5296761539769358, (4, 2) = 11.66918438963109, (5, 1) = .7057258717088374, (5, 2) = 10.944263938053027, (6, 1) = .8727045160581467, (6, 2) = 10.12940781981862, (7, 1) = 1.0293675300859457, (7, 2) = 9.286027153268176, (8, 1) = 1.1758543805831463, (8, 2) = 8.457430747175952, (9, 1) = 1.3127197264585053, (9, 2) = 7.671647779381384, (10, 1) = 1.4404597517069202, (10, 2) = 6.945889507639623, (11, 1) = 1.5602083118771684, (11, 2) = 6.28529164237151, (12, 1) = 1.6729138904127339, (12, 2) = 5.690377357385402, (13, 1) = 1.7796245072672423, (13, 2) = 5.157666928630136, (14, 1) = 1.8817159269340347, (14, 2) = 4.680400525330798, (15, 1) = 1.9802286821314778, (15, 2) = 4.2529853303192855, (16, 1) = 2.0768238867507085, (16, 2) = 3.8675310705673276, (17, 1) = 2.173068367774721, (17, 2) = 3.5178917521371798, (18, 1) = 2.270890048473742, (18, 2) = 3.1984139900960797, (19, 1) = 2.3732345057338295, (19, 2) = 2.9030738541180425, (20, 1) = 2.4832304722226684, (20, 2) = 2.6296687269685846, (21, 1) = 2.607217350093064, (21, 2) = 2.3751465147467514, (22, 1) = 2.7547856154152726, (22, 2) = 2.144274550259719, (23, 1) = 2.939616093527199, (23, 2) = 1.961813066469559, (24, 1) = 3.162337834655888, (24, 2) = 1.8947028342219836, (25, 1) = 3.3794261487929442, (25, 2) = 1.9881175251450118, (26, 1) = 3.5551825581028798, (26, 2) = 2.1804710450280265, (27, 1) = 3.69665130286631, (27, 2) = 2.41371269610685, (28, 1) = 3.8169541259171704, (28, 2) = 2.6689707564501304, (29, 1) = 3.9243694765971413, (29, 2) = 2.9422812417392152, (30, 1) = 4.024767234931778, (30, 2) = 3.237123523436467, (31, 1) = 4.121259324516239, (31, 2) = 3.5565852574603505, (32, 1) = 4.216529428559537, (32, 2) = 3.9064929391469123, (33, 1) = 4.312457356149003, (33, 2) = 4.292723456572624, (34, 1) = 4.410412203730131, (34, 2) = 4.720850296041311, (35, 1) = 4.511931211523759, (35, 2) = 5.198255733142234, (36, 1) = 4.618090131180544, (36, 2) = 5.730648052338743, (37, 1) = 4.729865319802914, (37, 2) = 6.322579877258892, (38, 1) = 4.848375884647389, (38, 2) = 6.977624814470957, (39, 1) = 4.974387707359408, (39, 2) = 7.694131606716463, (40, 1) = 5.108607237299307, (40, 2) = 8.464739698798486, (41, 1) = 5.251606222026279, (41, 2) = 9.273734329435953, (42, 1) = 5.403405586640311, (42, 2) = 10.092710388426758, (43, 1) = 5.563684034528413, (43, 2) = 10.88121631033313, (44, 1) = 5.731806698053641, (44, 2) = 11.588598491150355, (45, 1) = 5.905935538945928, (45, 2) = 12.156134835319502, (46, 1) = 6.0836133491883215, (46, 2) = 12.530876289186718, (47, 1) = 6.262647078633722, (47, 2) = 12.678160168039451, (48, 1) = 6.439921199996006, (48, 2) = 12.587708080348033, (49, 1) = 6.612315033049238, (49, 2) = 12.2791039631489, (50, 1) = 6.777770156025198, (50, 2) = 11.79378728978802, (51, 1) = 6.934742040084409, (51, 2) = 11.183836634124887, (52, 1) = 7.0820078226929555, (52, 2) = 10.502833198232775, (53, 1) = 7.219149840143932, (53, 2) = 9.796155822525524, (54, 1) = 7.34648415514996, (54, 2) = 9.096522907410908, (55, 1) = 7.464338781940853, (55, 2) = 8.427075143053173, (56, 1) = 7.573265028033481, (56, 2) = 7.801521511288004, (57, 1) = 7.67403328840954, (57, 2) = 7.22588851589702, (58, 1) = 7.767374215067717, (58, 2) = 6.701837954309735, (59, 1) = 7.85398163397448, (59, 2) = 6.228168238234758}, datatype = float[8], order = C_order); YP := Matrix(59, 2, {(1, 1) = 12.679746440910824, (1, 2) = -.0, (2, 1) = 12.567764844010108, (2, 2) = -16.198974726732907, (3, 1) = 12.226735465168346, (3, 2) = -31.019959257656904, (4, 1) = 11.66918438963109, (4, 2) = -42.61556998298658, (5, 1) = 10.944263938053027, (5, 2) = -49.758388561480395, (6, 1) = 10.12940781981862, (6, 2) = -52.428967468700804, (7, 1) = 9.286027153268176, (7, 2) = -51.59782241363348, (8, 1) = 8.457430747175952, (8, 2) = -48.43560278055239, (9, 1) = 7.671647779381384, (9, 2) = -43.97106435599, (10, 1) = 6.945889507639623, (10, 2) = -38.985423814257885, (11, 1) = 6.28529164237151, (11, 2) = -33.97931379570107, (12, 1) = 5.690377357385402, (12, 2) = -29.25037640167964, (13, 1) = 5.157666928630136, (13, 2) = -24.944278873434246, (14, 1) = 4.680400525330798, (14, 2) = -21.099971062135964, (15, 1) = 4.2529853303192855, (15, 2) = -17.71654087434397, (16, 1) = 3.8675310705673276, (16, 2) = -14.74668416472017, (17, 1) = 3.5178917521371798, (17, 2) = -12.14221532932914, (18, 1) = 3.1984139900960797, (18, 2) = -9.851136892783515, (19, 1) = 2.9030738541180425, (19, 2) = -7.815047340827251, (20, 1) = 2.6296687269685846, (20, 2) = -5.996847641998252, (21, 1) = 2.3751465147467514, (21, 2) = -4.340048329343317, (22, 1) = 2.144274550259719, (22, 2) = -2.7978073376293024, (23, 1) = 1.961813066469559, (23, 2) = -1.320655805545758, (24, 1) = 1.8947028342219836, (24, 2) = .13039610824419176, (25, 1) = 1.9881175251450118, (25, 2) = 1.5787914768274436, (26, 1) = 2.1804710450280265, (26, 2) = 3.0488939854293733, (27, 1) = 2.41371269610685, (27, 2) = 4.590848716067672, (28, 1) = 2.6689707564501304, (28, 2) = 6.254836311594842, (29, 1) = 2.9422812417392152, (29, 2) = 8.080806575409401, (30, 1) = 3.237123523436467, (30, 2) = 10.12389716847413, (31, 1) = 3.5565852574603505, (31, 2) = 12.425685648274625, (32, 1) = 3.9064929391469123, (32, 2) = 15.042554769209124, (33, 1) = 4.292723456572624, (33, 2) = 18.027628684062055, (34, 1) = 4.720850296041311, (34, 2) = 21.42377820734578, (35, 1) = 5.198255733142234, (35, 2) = 25.27273224998513, (36, 1) = 5.730648052338743, (36, 2) = 29.574358687095494, (37, 1) = 6.322579877258892, (37, 2) = 34.27026010854073, (38, 1) = 6.977624814470957, (38, 2) = 39.21645062769387, (39, 1) = 7.694131606716463, (39, 2) = 44.11406880450838, (40, 1) = 8.464739698798486, (40, 2) = 48.471285191096335, (41, 1) = 9.273734329435953, (41, 2) = 51.56564318350719, (42, 1) = 10.092710388426758, (42, 2) = 52.45733113498812, (43, 1) = 10.88121631033313, (43, 2) = 50.12848157993341, (44, 1) = 11.588598491150355, (44, 2) = 43.73470437900077, (45, 1) = 12.156134835319502, (45, 2) = 33.00620482181849, (46, 1) = 12.530876289186718, (46, 2) = 18.57982197925053, (47, 1) = 12.678160168039451, (47, 2) = 1.9582711648426194, (48, 1) = 12.587708080348033, (48, 2) = -14.727409356953261, (49, 1) = 12.2791039631489, (49, 2) = -29.39558090446206, (50, 1) = 11.79378728978802, (50, 2) = -40.669372493045, (51, 1) = 11.183836634124887, (51, 2) = -48.03607321421275, (52, 1) = 10.502833198232775, (52, 2) = -51.72939943874306, (53, 1) = 9.796155822525524, (53, 2) = -52.452900134134595, (54, 1) = 9.096522907410908, (54, 2) = -51.04757024201683, (55, 1) = 8.427075143053173, (55, 2) = -48.28611154332724, (56, 1) = 7.801521511288004, (56, 2) = -44.78604220693657, (57, 1) = 7.22588851589702, (57, 2) = -40.986432776247455, (58, 1) = 6.701837954309735, (58, 2) = -37.176690679157566, (59, 1) = 6.228168238234758, (59, 2) = -33.53206368300497}, datatype = float[8], order = C_order); errproc := proc (x_bvp) local outpoint, X, Y, yout, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; Digits := 15; outpoint := evalf(x_bvp); X := Vector(59, {(1) = .0, (2) = 0.13680812436809657e-1, (3) = 0.27971051145644167e-1, (4) = 0.4296023135974479e-1, (5) = 0.5851786182852681e-1, (6) = 0.7435973609701907e-1, (7) = 0.9049963296106847e-1, (8) = .1070194360291177, (9) = .12400404122395811, (10) = .14149911102616744, (11) = .15962080170244286, (12) = .17846661163773977, (13) = .1981659899247577, (14) = .2189484735830971, (15) = .2410342106109193, (16) = .2648591917099649, (17) = .29096256417430655, (18) = .3201405168424954, (19) = .3537506263213988, (20) = .39359925436482457, (21) = .44328213017143814, (22) = .508829664760916, (23) = .5993459008179163, (24) = .7156934169226271, (25) = .8282966673508598, (26) = .9130409367878883, (27) = .9748398788122531, (28) = 1.0223008015052495, (29) = 1.0606661846616776, (30) = 1.0932188074676115, (31) = 1.1216707736458404, (32) = 1.1472399927159063, (33) = 1.1706725764522086, (34) = 1.1924372702961963, (35) = 1.2129338175389266, (36) = 1.2323858241405192, (37) = 1.250955152949817, (38) = 1.268795762648431, (39) = 1.2859897085262963, (40) = 1.3026146566339305, (41) = 1.3187452976919192, (42) = 1.3344237627757964, (43) = 1.3497030677053556, (44) = 1.3646567912885574, (45) = 1.3793073520205892, (46) = 1.3936816310415334, (47) = 1.407863413800478, (48) = 1.421874455055364, (49) = 1.4357209480594326, (50) = 1.449452395680341, (51) = 1.463105461314161, (52) = 1.4766814720936998, (53) = 1.4901926102416252, (54) = 1.5036746366401115, (55) = 1.5171303716739315, (56) = 1.5305608572874234, (57) = 1.5439797161303026, (58) = 1.5573915356105725, (59) = 1.5707963267949}, datatype = float[8], order = C_order); Y := Matrix(59, 2, {(1, 1) = .0, (1, 2) = -0.2040971830865995e-12, (2, 1) = -0.2515085127931946e-15, (2, 2) = -0.2182240868305703e-12, (3, 1) = -0.3034400215680231e-14, (3, 2) = -0.24629572182126297e-12, (4, 1) = -0.70293294530316345e-14, (4, 2) = -0.27355561891671303e-12, (5, 1) = -0.11642509457266535e-13, (5, 2) = -0.2921725988376451e-12, (6, 1) = -0.14891698174826332e-13, (6, 2) = -0.27332966679506926e-12, (7, 1) = -0.2244158402930937e-13, (7, 2) = -0.21555448086863683e-12, (8, 1) = -0.23830070405255755e-13, (8, 2) = -0.14452163468605652e-12, (9, 1) = -0.28784241689197694e-13, (9, 2) = -0.8936429016769738e-13, (10, 1) = -0.2870763992607896e-13, (10, 2) = -0.7015153701410498e-13, (11, 1) = -0.29805232120338505e-13, (11, 2) = -0.20455226198960395e-13, (12, 1) = -0.3071869736898188e-13, (12, 2) = -0.949608712395301e-14, (13, 1) = -0.33085750279284234e-13, (13, 2) = -0.9853725522439467e-14, (14, 1) = -0.3024025650769599e-13, (14, 2) = 0.55343272382320285e-14, (15, 1) = -0.2989762453456761e-13, (15, 2) = 0.9484091577415516e-14, (16, 1) = -0.3126420836975352e-13, (16, 2) = 0.10993914919372806e-15, (17, 1) = -0.2904952409517121e-13, (17, 2) = 0.7741875869166384e-14, (18, 1) = -0.26389405390354672e-13, (18, 2) = -0.39736138956257974e-14, (19, 1) = -0.3066642976676729e-13, (19, 2) = -0.8841791351636723e-14, (20, 1) = -0.3381874764737987e-13, (20, 2) = -0.10474029579197825e-13, (21, 1) = -0.3076950762715862e-13, (21, 2) = -0.14630357321113296e-13, (22, 1) = -0.25742990878046857e-13, (22, 2) = -0.32013707096004785e-13, (23, 1) = -0.26034531106847924e-14, (23, 2) = -0.17882398235769704e-13, (24, 1) = -0.24294629294277565e-13, (24, 2) = 0.6033751098446911e-13, (25, 1) = -0.10212795741481291e-13, (25, 2) = -0.5409909790245115e-14, (26, 1) = 0.65234775286404595e-14, (26, 2) = 0.10506976038853487e-14, (27, 1) = 0.15145921627767115e-13, (27, 2) = 0.11062582034309036e-13, (28, 1) = 0.17727034515154246e-13, (28, 2) = 0.1392374003706416e-13, (29, 1) = 0.20190304444405087e-13, (29, 2) = 0.24077662756883795e-13, (30, 1) = 0.15567756982704383e-13, (30, 2) = 0.3220356728033926e-13, (31, 1) = 0.2049236289955625e-13, (31, 2) = 0.4582315826902611e-13, (32, 1) = 0.2093779309445625e-13, (32, 2) = 0.50847503530540665e-13, (33, 1) = 0.34656722670579105e-13, (33, 2) = 0.65791408284367e-13, (34, 1) = 0.1414627403891834e-13, (34, 2) = 0.7434970442839986e-13, (35, 1) = 0.1430197385087178e-13, (35, 2) = 0.807995248637132e-13, (36, 1) = 0.2674246752632126e-13, (36, 2) = 0.8238316242997303e-13, (37, 1) = 0.3749521089102351e-13, (37, 2) = 0.9551972331079688e-13, (38, 1) = 0.43798173157104124e-13, (38, 2) = 0.8882243056529444e-13, (39, 1) = 0.34765247192333954e-13, (39, 2) = 0.7174268116999525e-13, (40, 1) = 0.3492718539460031e-13, (40, 2) = 0.25074509668472094e-13, (41, 1) = 0.3268343950774218e-13, (41, 2) = -0.6182099171300564e-13, (42, 1) = 0.7615127467050659e-14, (42, 2) = -0.1854246486047524e-12, (43, 1) = 0.20280258788197414e-13, (43, 2) = -0.2612655964015281e-12, (44, 1) = 0.5268977580072408e-14, (44, 2) = -0.3058713868413555e-12, (45, 1) = 0.8243135377351008e-14, (45, 2) = -0.3000324869932838e-12, (46, 1) = 0.9651203882669847e-14, (46, 2) = -0.23958986839330786e-12, (47, 1) = -0.15172603499122148e-13, (47, 2) = -0.14114952424844551e-12, (48, 1) = 0.6415293117671377e-14, (48, 2) = -0.3574626525598719e-13, (49, 1) = 0.11771473310921094e-13, (49, 2) = 0.12758808412240534e-13, (50, 1) = 0.350361321759921e-14, (50, 2) = 0.18960251337597162e-13, (51, 1) = -0.402567915071639e-14, (51, 2) = -0.42598728779319296e-13, (52, 1) = 0.14493706543893737e-13, (52, 2) = -0.4380811741884113e-13, (53, 1) = 0.19531436506149837e-13, (53, 2) = -0.8490114617635297e-13, (54, 1) = 0.4784180312117899e-14, (54, 2) = -0.10238102255674353e-12, (55, 1) = 0.13186433513432542e-13, (55, 2) = -0.12142340951879697e-12, (56, 1) = 0.68589188913736246e-14, (56, 2) = -0.10734720880680753e-12, (57, 1) = 0.4428481028761533e-14, (57, 2) = -0.9866461770792047e-13, (58, 1) = 0.7387414481358141e-14, (58, 2) = -0.9166293971010342e-13, (59, 1) = .0, (59, 2) = -0.731647525953621e-13}, datatype = float[8], order = C_order); if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "right" then return X[59] elif outpoint = "order" then return 10 elif outpoint = "error" then return HFloat(3.058713868413555e-13) elif outpoint = "errorproc" then error "this is already the error procedure" elif outpoint = "rawdata" then return [2, 59, [v(u), diff(v(u), u)], X, Y] else return ('procname')(x_bvp) end if end if; if outpoint < X[1] or X[59] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[59] end if; V := array([1 = 4, 2 = 0]); if Digits <= trunc(evalhf(Digits)) then L := Vector(4, 'datatype' = 'float'[8]); yout := Vector(2, 'datatype' = 'float'[8]); evalhf(`dsolve/numeric/lagrange`(59, 2, X, Y, outpoint, var(yout), var(L), var(V))) else L := Vector(4, 'datatype' = 'sfloat'); yout := Vector(2, 'datatype' = 'sfloat'); `dsolve/numeric/lagrange`(59, 2, X, Y, outpoint, yout, L, V) end if; [u = outpoint, seq('[v(u), diff(v(u), u)]'[i] = yout[i], i = 1 .. 2)] end proc; if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "method" then return "bvp" elif outpoint = "right" then return X[59] elif outpoint = "order" then return 10 elif outpoint = "error" then return HFloat(3.058713868413555e-13) elif outpoint = "errorproc" then return eval(errproc) elif outpoint = "rawdata" then return [2, 59, "depnames", X, Y, YP] else error "non-numeric value" end if end if; if outpoint < X[1] or X[59] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[59] end if; if Digits <= trunc(evalhf(Digits)) and (_EnvInFsolve <> true or _EnvDSNumericSaveDigits <= trunc(evalhf(Digits))) then V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false) ] ); L := Matrix(7, 2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0, (3, 1) = .0, (3, 2) = .0, (4, 1) = .0, (4, 2) = .0, (5, 1) = .0, (5, 2) = .0, (6, 1) = .0, (6, 2) = .0, (7, 1) = .0, (7, 2) = .0}, datatype = float[8], order = C_order); yout := Vector(2, {(1) = .0, (2) = .0}, datatype = float[8]); evalhf(`dsolve/numeric/hermite`(59, 2, X, Y, YP, outpoint, var(yout), var(L), var(V))) else if _EnvInFsolve = true then Digits := _EnvDSNumericSaveDigits end if; V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false) ] ); L := Matrix(7, 2, {(1, 1) = 0., (1, 2) = 0., (2, 1) = 0., (2, 2) = 0., (3, 1) = 0., (3, 2) = 0., (4, 1) = 0., (4, 2) = 0., (5, 1) = 0., (5, 2) = 0., (6, 1) = 0., (6, 2) = 0., (7, 1) = 0., (7, 2) = 0.}, order = C_order); yout := Vector(2, {(1) = 0., (2) = 0.}); `dsolve/numeric/hermite`(59, 2, X, Y, YP, outpoint, yout, L, V) end if; [outpoint, seq(yout[i], i = 1 .. 2)] end proc, (2) = Array(1..3, {(1) = 36893628625353841228, (2) = 36893628625353841404, (3) = 36893628625353841580}), (3) = [u, v(u), diff(v(u), u)], (4) = 0}); solnproc := data[1]; if not type(outpoint, 'numeric') then if outpoint = "solnprocedure" then return eval(solnproc) elif member(outpoint, ["start", "left", "right", "errorproc", "rawdata", "order", "error"]) then return solnproc(u) elif outpoint = "sysvars" then return data[3] elif procname <> unknown then return ('procname')(u) else `D(v)` := pointto(data[2][3]); return ('`D(v)`')(u) end if end if; try res := solnproc(outpoint); res[3] catch: error end try end proc]

> 

display(
        torus,
        tubeplot([seq(eval(x(u, v(u)), dsol1))], u=0..Pi/2, color=red, radius=0.1),
        tubeplot([seq(eval(x(u, v(u)), dsol3))], u=0..Pi/2, color=cyan, radius=0.1),
        pointplot3d(eval([x(0,v(0)),x(Pi/2,v(Pi/2))], dsol3), symbol=solidsphere, symbolsize=20, color="Orange"),
lightmodel=light4, thickness=5, orientation=[-65,45,0], size=[700,400],
style=surface, axes=none, viewpoint=circleright);

> 

 

> 

 

>

 

Download geodesics-between-two-points.mw

First 9 10 11 12 13 14 15 Last Page 11 of 58