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Dear experts;

How can I solve this problem with maple?

restart:


 X[3](0):=6.3096*10^9;
 c:=0.67;
 d:=3.7877*10^(-8);
 delta:=3.259*d;
 lambda:=(2/3)*10^8*d;
 R[0]:=1.33;
 p:=(c*X[3](0)*delta*R[0])/(lambda*(R[0]-1));
beta:=(d*delta*c*R[0])/(lambda*p);

ode:=diff(x[1](t), t)=(lambda-d*x[1](t)-(1-beta*x[1](t)*x[3](t)*(psi[1](t)-psi[2](t))/A[1])*beta*x[1](t)*x[3](t)),
 diff(x[2](t), t) =((1-beta*x[1](t)*x[3](t)*(psi[1](t)-psi[2](t))/A[1])*beta*x[1](t)*x[3](t)-delta*x[2](t)),
 diff(x[3](t), t) =((1+psi[3](t)*p*x[2](t)/A[2])*p*x[2](t)-c*x[3](t)),diff(psi[1](t), t) =-1+1/A[1]*beta^2*x[1](t)*(x[3](t))^2*(psi[1](t)-psi[2](t))^2-psi[1](t)*(-d+beta^2*(x[3](t))^2*(psi[1](t)-psi[2](t))/A[1]*x[1](t)-(1-beta*x[1](t)*x[3](t)*(psi[1](t)-psi[2](t))/A[1])*beta*x[3](t))-psi[2](t)*(-beta^2*(x[3](t))^2*(psi[1](t)-psi[2](t))/A[1]*x[1](t)+(1-beta*x[1](t)*x[3](t)*(psi[1](t)-psi[2](t))/A[1])*beta*x[3](t)),
 diff(psi[2](t), t) =1/A[2]*psi[3](t)^2*p^2*x[2](t)+psi[2](t)*delta-psi[3](t)*(psi[3](t)*p^2/A[2]*x[2](t)+(1+psi[3](t)*p*x[2](t)/A[2])*p),
 diff(psi[3](t), t) = 1/A[1]*beta^2*(x[1](t))^2*x[3](t)*(psi[1](t)-psi[2](t))^2-psi[1](t)*(beta^2*(x[1](t))^2*(psi[1](t)-psi[2](t))/A[1]*x[3](t)-(1-beta*x[1](t)*x[3](t)*(psi[1](t)-psi[2](t))/A[1])*beta*x[1](t))-psi[2](t)*(-beta^2*(x[1](t))^2*(psi[1](t)-psi[2](t))/A[1]*x[3](t)+(1-beta*x[1](t)*x[3](t)*(psi[1](t)-psi[2](t))/A[1])*beta*x[1](t))+psi[3](t)*c;

ics := x[1](0)=5.5556*10^7, x[2](0)=1.1111*10^7,x[3](0)=6.3096*10^9,psi[1](100)=0,psi[2](100)=0,psi[3](100)=0;

dsolve([ode, ics],numeric);?????????????????????????

Please help me

ode.mws

Minimize doesn't work with dsolve porcedure?

experiment_real.mw

tr := proc (x, y)::integer; tr := x+y; result := x^(2+y) end proc

Warning, `result` is implicitly declared local to procedure `tr`

 

tr(5, 5)

78125

(1)

with(Optimization); Minimize(tr(x, y), x = 0 .. 1000, y = 1 .. 1, initialpoint = {x = 25, y = 1})

[0.117556065072605623e-15, [x = HFloat(4.898709434833346e-6), y = HFloat(1.0)]]

(2)

xxx := 97.39391293; yyy := -1.588898710

-1.588898710

(3)

xx := 100;

3

(4)

trool := proc (leng, alpha)::integer; global psi, zx, zy, xx, yy, xxx, yyy, sa, ca, ps, Vx, Vy, vx, vy, ode, ics, XX, YY, trool, G, str, start, ds; sa := evalf(sin(alpha)); ca := evalf(cos(alpha)); ps := evalf(evalc(Im(evalc(str*(x+I*y)-((1/2)*I)*G*ln(x+I*y-start)/Pi)))); psi := ps; xxx := evalf(xx+leng*ca); yyy := evalf(yy+leng*sa); Vx := diff(psi, y); Vy := -(diff(psi, x)); vx := Re(evalf(subs(x = xxx, y = yyy, subs(vvx = Vx, vvx)))); vy := Re(evalf(subs(x = xxx, y = yyy, subs(vvy = Vy, vvy)))); proc (X) options operator, arrow; X(t) end proc; proc (Y) options operator, arrow; Y(t) end proc; zx := proc (t) options operator, arrow; evalf(subs(x = X(t), y = Y(t), subs(vvx = Vx, vvx))) end proc; zy := proc (t) options operator, arrow; evalf(subs(x = X(t), y = Y(t), subs(vvy = Vy, vvy))) end proc; ode := diff(X(t), t) = zx(t), diff(Y(t), t) = zy(t); ics := X(0) = xxx, Y(0) = yyy; ds := dsolve([ode, ics], type = numeric, [X(t), Y(t)], method = rkf45, maxfun = 0, output = listprocedure, abserr = 0.1e-3, relerr = 0.1e-3, minstep = 0.1e-1); XX := rhs(ds[2]); YY := rhs(ds[3]); trool := XX(0.1e-3) end proc:

with(Optimization); Minimize(trool(alpha, leng), assume = nonnegative, alpha = 0 .. 2*Pi, leng = .2 .. 2, iterationlimit = 1000, initialpoint = {alpha = 1, leng = 1})

Error, (in XX) parameter 'alpha' must be assigned a numeric value before obtaining a solution

 

alpha = 0 .. 2*Pi, leng = .2 .. 2, output = solutionmodule

alpha := 1; leng := 1; XX(10)

HFloat(100.54666738117751)

(5)

``

trool(1, 11)

HFloat(100.00711298362239)

(6)

psi

3.*y-11.93662073*ln((x-100.)^2+y^2)

(7)

``

 

Download experiment_real.mw

with trool procedure minimize dosent work .... and its make me realy sad, couse i need to optimize alpha and leng in other (big one) porcedure with same dsolve.

get this errors:
"Warning, The use of global variables in numerical ODE problems is deprecated, and will be removed in a future release. Use the 'parameters' argument instead (see ?dsolve,numeric,parameters)"
"Error, (in XX) parameter 'alpha' must be assigned a numeric value before obtaining a solution"

I have following expression

f:=t->((1/8)*s^2*sinh(4*t)+t+(1/2)*s^2*t+s*sinh(2*t))/(1+s*cosh(2*t))

which is 1 solution of the ODE

ode2 := -(diff(y(t), t, t))+(4-12/(1+s*cosh(2*t))+(8*(-s^2+1))/(1+s*cosh(2*t))^2)*y(t) = 0

Now I wanted to construct 2 linear independent solutions via:

f1:=f(t_b-t)

f2:=f(t-t_a)

and calculate the Wronskian:

with(LinearAlgebra); with(VectorCalculus)

Determinant(Wronskian([f(t_b-t), f(t-t_a)], t))

Since I know these functions are solutions of the second order ODE which does not contain any first order derivative the Wronskian should be a constant. Unfortunately Maple has a hard time to simplify it since the epxression is a little big. Is it my fault or has anyone an idea what to do?

I have a problem in excuting this differential equation in maple it takes a long time but yet no result.

> restart;


> Delta:= epsilon[2]-epsilon[1];

> epsilon[y] := epsilon[2]-(1/4)*Delta*(1-tanh(a*y))^2;

 > z:= tanh(a*y) ;

 > ODE[4]:= diff(Y(y),y,y)- ( a/2* Delta *(1-z)*(z^2-1))/(epsilon[2]- Delta*(1-z)/4)* diff(Y(y),y)-( beta^2+ mu[0]*epsilon[y]*omega^2)*Y(y) = 0;

> dsolve(ODE[4],Y(y));

does this always occur or i do have problem with my version of maple 15, 7 and 16.

Thank you, looking forward for your answers.

 

I am trying to solve a nonlinear second order ODE with a parameter to be determined. I can set up most of the problem but I am having trouble trying to tell the computer the following boundary condition,

 

diff(f(x),x) = -K on f(x)=0.

 

(K will be inputted and is not to be solved for) As i said before, the other boundary conditions are fine and the numerical solution works if i use different boundary conditions. 

 

For other boundary conditions (for example df/dx = 0 at x=0) I write in the form 

D(f)(0)=0

 

I hope this makes sense and someone has a solution. Thanks in advance.

 

Matt

Given the following system of first order ODE,

dx/dt=0.2x(1-0.5 x)-(1.5 xy)/(1+0.116 x),

dy/dt=(1.3 xy)/(1+0.1x)-0.8y.

 

Draw a DEplot (for t from 0 to 50) and indicate the particular

trajectory with the initial conditions x(0)=1,y(0)=2. If I

switched to forward Euler method,what would the DE plot look

like then? Is it possible to make the plot made by the

forward Euler method look close to the one which used the

default method?

As it says in the title, I would like to solve the following ODE numerically using forward Euler method, without using the Student Package.

 

(dy(t))/(dt)=t(1-0.3t)-(ty)(1+0.6t)

with initial condition y(0)=1. I want to solve it for up to t=1, and then plot both the solution by Euler's method and the solution by "dsolve" on the same graph so I can compare them.

 

Also, can I make a separate DEplot with t extending to 5?

 

Thanks in advance.

Following previous question at

http://www.mapleprimes.com/questions/149581-Improve-Algorithm-Dsolve

and also

http://www.mapleprimes.com/questions/149243-BVP-With-Constraining-Integrals

I wrote the following code

***********************

restart:

gama1:=0:


phi0:=0.00789:


rhocu:=2/(1-zet^2)*int((1-eta)*rho(eta)*c(eta)*u(eta),eta=0..1-zet):

eq1:=diff(u(eta),eta,eta)+1/(mu(eta)/mu1[w])+((1/(eta-1)+1/mu(eta)*(mu_phi*diff(phi(eta),eta)))*diff(u(eta),eta)):
eq2:=diff(T(eta),eta,eta)+1/(k(eta)/k1[w])*(2/(1-zet^2)*rho(eta)*c(eta)*u(eta)/(p2*10000)+( (a[k1]+2*b[k1]*phi(eta))/(1+a[k1]*phi1[w]+b[k1]*phi1[w]^2)*diff(phi(eta),eta)-k(eta)/k1[w]/(1-eta)*diff(T(eta),eta) )):
eq3:=diff(phi(eta),eta)-phi(eta)/(N[bt]*(1-gama1*T(eta))^2)*diff(T(eta),eta):
mu:=unapply(mu1[bf]*(1+a[mu1]*phi(eta)+b[mu1]*phi(eta)^2),eta):
k:=unapply(k1[bf]*(1+a[k1]*phi(eta)+b[k1]*phi(eta)^2),eta):
rhop:=3880:
rhobf:=998.2:
cp:=773:
cbf:=4182:
rho:=unapply(  phi(eta)*rhop+(1-phi(eta))*rhobf ,eta):
c:=unapply(  (phi(eta)*rhop*cp+(1-phi(eta))*rhobf*cbf )/rho(eta) ,eta):
mu_phi:=mu1[bf]*(a[mu1]+2*b[mu1]*phi(eta)):

a[mu1]:=39.11:
b[mu1]:=533.9:
mu1[bf]:=9.93/10000:
a[k1]:=7.47:
b[k1]:=0:
k1[bf]:=0.597:
zet:=0.5:
#phi(0):=1:
#u(0):=0:
phi1[w]:=phi0:
N[bt]:=0.2:
mu1[w]:=mu(0):
k1[w]:=k(0):

eq1:=subs(phi(0)=phi0,u(0)=0,eq1):
eq2:=subs(phi(0)=phi0,u(0)=0,eq2):
eq3:=subs(phi(0)=phi0,u(0)=0,eq3):

p:=proc(pp2) global res,F0,F1,F2:
if not type([pp2],list(numeric)) then return 'procname(_passed)' end if:
res := dsolve({eq1=0,subs(p2=pp2,eq2)=0,eq3=0,u(0)=0,u(1-zet)=0,phi(0)=phi0,T(0)=0,D(T)(0)=1}, numeric,output=listprocedure):
F0,F1,F2:=op(subs(res,[u(eta),phi(eta),T(eta)])):
evalf(2/(1-zet^2)*Int((1-eta)*(F1(eta)*rhop+(1-F1(eta))*rhobf)*( F1(eta)*rhop*cp+(1-F1(eta))*rhobf*cbf )/(F1(eta)*rhop+(1-F1(eta))*rhobf)*F0(eta),eta=0..1-zet))-pp2*10000:
end proc:


s1:=Student:-NumericalAnalysis:-Secant(p(pp2),pp2=[6,7],tolerance=1e-6);

                   HFloat(6.600456858832996)

p2:=%:



ruu:=evalf(2/(1-zet^2)*(Int((1-eta)*F0(eta),eta=0..1-zet))):
phb:=evalf(2/(1-zet^2)*(Int((1-eta)*F0(eta)*F1(eta),eta=0..1-zet))) / evalf(2/(1-zet^2)*(Int((1-eta)*F0(eta),eta=0..1-zet))) :
TTb:=evalf(2/(1-zet^2)*(Int((1-eta)*F2(eta),eta=0..1-zet))):
rhouu:=evalf(2/(1-zet^2)*(Int((1-eta)*(F1(eta)*rhop+(1-F1(eta))*rhobf)*F0(eta),eta=0..1-zet))):
with(plots):
res(parameters=[R0,R1]):
odeplot(res,[[eta,u(eta)/ruu],[eta,phi(eta)/phb],[eta,T(eta)/TTb]],0..zet);

 

*************************************

as you can see at the second line of the code, the value of phi0:=0.00789. however, I want to modify the code in a way that phi0 is calculated with the following addition constraint

evalf(2/(1-zet^2)*(Int((1-eta)*F0(eta)*F1(eta),eta=0..1-zet))) / evalf(2/(1-zet^2)*(Int((1-eta)*F0(eta),eta=0..1-zet)))-0.02=0

I would be most grateful if you could help me in this problem.

Thanks for your attention in advance

Amir

Here is a screen of the original question  http://www.mapleprimes.com/ViewTemp.ashx?f=21095_1385385286/screen25.11.13.docx

 My advice to the questioner is to visit a psychiatrist ASAP.

Markiyan Hirnyk

 

Thank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank youThank you

So i have to plot 3 ODEs. How would I go about plotting dx/dt=10(y-x) AND dy/dt=x(28-x)-y AND dz/dt= xy-8z/3. Any help at all would be appreciated.

The question is to solve

x^2*(diff(y(x), x, x))+x^3*(diff(y(x), x))+(x^2-2)*y(x) = 0

and evaluate it where y(1) = 1 and y(2) = 2 to find y(3).

 

When I do the equation,

 

ODE := x^2*(diff(y(x), x, x))+x^3*(diff(y(x), x))+(x^2-2)*y(x) = 0

dsolve(ODE) y(x) = _C1/x+_C2*(-sqrt(Pi)*sqrt(2)*erf((1/2)*sqrt(2)*x)+2*x*exp(-(1/2)*x^2))/x

 

I have gotten an erf. Is this correct?

Hi -

I'm trying to use Maple to integrate an ODE numerically, but to output the integration steps as I go (in an array, or similar). None of the dsolve/numeric output parameters appear to offer this (ie

x=0.00 f = whatever,

x = 0.05 f = whatever2,

x = 0.10 f = whatever3)

Is there a standard feature in Maple that I'm missing, or would I have to force the output from the dsolve procedure in some way, please?

I have 2nd order nonlinear ode I try to solve with Runge Kutta 4th order method in maple but all I get from the outcome was 1 and 0.This is the equation:theta_ode1.mw . How do I do it Or how do I write the code to solve it with maple using  Runge Kutta 4th order method?

I have 2nd order nonlinear ode I try to solve with Runge Kutta 4th order method in maple but all I get from the out is 1 and 0.This is the equation: theta_ode.mw . How do I do it Or how do I write the code to solve it with maple using  Runge Kutta 4th order method?

Kepler_Gravitatio.mw

restart; Digits := 64; assume(r > 0); epsilon := 0.167e-1; AE := 149597870700; mu := 1; v_T := 30290; r_0 := .983*AE; M_0 := 0.; GM := 132712440018*10^9; L_0 := sqrt(GM*mu^2*r_0*(1+epsilon)); d := 86400; ode := diff(r(t), t, t)+(-(L_0+M_0*t)^2/(mu^2*r(t)^3)+GM/r(t)^2)*d^2 = 0

diff(diff(r(t), t), t)-148119034496962719477804289013145600000000.0000000000000000000000/r(t)^3+990693056236769280000000000000/r(t)^2 = 0

(1)

sol := dsolve({ode, r(0) = r_0, (D(r))(0) = 0}, r(t), numeric)

proc (x_rkf45) local _res, _dat, _vars, _solnproc, _xout, _ndsol, _pars, _n, _i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; if 1 < nargs then error "invalid input: too many arguments" end if; _EnvDSNumericSaveDigits := Digits; Digits := 64; if _EnvInFsolve = true then _xout := evalf[_EnvDSNumericSaveDigits](x_rkf45) else _xout := evalf(x_rkf45) end if; _dat := Array(1..4, {(1) = proc (_xin) local _xout, _dtbl, _dat, _vmap, _x0, _y0, _val, _dig, _n, _ne, _nd, _nv, _pars, _ini, _par, _i, _j, _k, _src; option `Copyright (c) 2002 by Waterloo Maple Inc. All rights reserved.`; table( [( "complex" ) = false ] ) _xout := _xin; _pars := []; _dtbl := array( 1 .. 4, [( 1 ) = (array( 1 .. 20, [( 1 ) = (datatype = sfloat, order = C_order, storage = rectangular), ( 2 ) = (datatype = sfloat, order = C_order, storage = rectangular), ( 3 ) = ([0, 0, 0, Array(1..0, 1..2, {}, order = C_order)]), ( 4 ) = (Array(1..53, {(1) = 2, (2) = 2, (3) = 0, (4) = 0, (5) = 0, (6) = 0, (7) = 1, (8) = 0, (9) = 0, (10) = 0, (11) = 0, (12) = 0, (13) = 0, (14) = 0, (15) = 0, (16) = 0, (17) = 0, (18) = 1, (19) = 30000, (20) = 0, (21) = 1, (22) = 1, (23) = 4, (24) = 0, (25) = 2, (26) = 64, (27) = 1, (28) = 0, (29) = 1, (30) = 3, (31) = 3, (32) = 0, (33) = 1, (34) = 0, (35) = 0, (36) = 0, (37) = 0, (38) = 0, (39) = 0, (40) = 0, (41) = 0, (42) = 0, (43) = 1, (44) = 0, (45) = 0, (46) = 0, (47) = 0, (48) = 0, (49) = 0, (50) = 50, (51) = 1, (52) = 0, (53) = 0}, datatype = integer[8])), ( 5 ) = (Array(1..28, {(1) = 0., (2) = 0.1e-5, (3) = 0., (4) = 0.500001e-63, (5) = 0., (6) = 0.6597695327722120288470717489122650270807917992426865066545877176e-8, (7) = 0., (8) = 0.1e-5, (9) = 0., (10) = 0., (11) = 0., (12) = 0., (13) = 1., (14) = 0., (15) = .49999999999999, (16) = 0., (17) = 1., (18) = 1., (19) = 0., (20) = 0., (21) = 1., (22) = 1., (23) = 0., (24) = 0., (25) = 0.1e-14, (26) = 0., (27) = 0., (28) = 0.}, order = C_order)), ( 6 ) = (Array(1..2, {(1) = 147054706898.100, (2) = 0.}, order = C_order)), ( 7 ) = ([Array(1..4, 1..7, {(1, 1) = 0., (1, 2) = 26., (1, 3) = 39., (1, 4) = 96., (1, 5) = 104., (1, 6) = 52., (1, 7) = 104., (2, 1) = 33440., (2, 2) = 0., (2, 3) = 146432., (2, 4) = 142805., (2, 5) = -50787., (2, 6) = 10260., (2, 7) = 282150., (3, 1) = 1045., (3, 2) = 0., (3, 3) = -11264., (3, 4) = -10985., (3, 5) = 7524., (3, 6) = 13680., (3, 7) = 376200., (4, 1) = 1629155., (4, 2) = 0., (4, 3) = 6769664., (4, 4) = 340535., (4, 5) = -101574., (4, 6) = -800280., (4, 7) = 13062500.}, order = C_order), Array(1..6, 1..6, {(1, 1) = 0., (1, 2) = 0., (1, 3) = 0., (1, 4) = 0., (1, 5) = 0., (1, 6) = 1., (2, 1) = 1., (2, 2) = 0., (2, 3) = 0., (2, 4) = 0., (2, 5) = 0., (2, 6) = 4., (3, 1) = 3., (3, 2) = 9., (3, 3) = 0., (3, 4) = 0., (3, 5) = 0., (3, 6) = 32., (4, 1) = 1932., (4, 2) = -7200., (4, 3) = 7296., (4, 4) = 0., (4, 5) = 0., (4, 6) = 2197., (5, 1) = 8341., (5, 2) = -32832., (5, 3) = 29440., (5, 4) = -845., (5, 5) = 0., (5, 6) = 4104., (6, 1) = -6080., (6, 2) = 41040., (6, 3) = -28352., (6, 4) = 9295., (6, 5) = -5643., (6, 6) = 20520.}, order = C_order), Array(1..6, {(1) = 0., (2) = .386, (3) = .210, (4) = .630, (5) = 1., (6) = 1.}, order = C_order), Array(1..6, {(1) = .25, (2) = -.1043, (3) = .1035, (4) = -0.362e-1, (5) = 0., (6) = 0.}, order = C_order), Array(1..6, 1..5, {(1, 1) = 0., (1, 2) = 0., (1, 3) = 0., (1, 4) = 0., (1, 5) = 0., (2, 1) = 1.544000000000000000000000000000000000000000000000000000000000000, (2, 2) = 0., (2, 3) = 0., (2, 4) = 0., (2, 5) = 0., (3, 1) = .9466785280815532861490926711419656433490789577360894528339016974, (3, 2) = .2557011698982581163688702568119981863592496590030907306661114512, (3, 3) = 0., (3, 4) = 0., (3, 5) = 0., (4, 1) = 3.314825187068488558786396153365059307030768681751721871755670487, (4, 2) = 2.896124015972123152514909728232581075175490361163408082084725849, (4, 3) = .9986419139977807257073047952984867573247485432986762804108143415, (4, 4) = 0., (4, 5) = 0., (5, 1) = 1.221224509226274823647507441203991684270141900788934785460003179, (5, 2) = 6.019134481287752905493183656137098641751723314760608648041583581, (5, 3) = 12.53708332932087457217734498320706156564427350079582065129891877, (5, 4) = -.6878860361058951356548892064135523542261590939207278512789580012, (5, 5) = 0., (6, 1) = 1.221224509226274823647507441203991684270141900788934785460003179, (6, 2) = 6.019134481287752905493183656137098641751723314760608648041583581, (6, 3) = 12.53708332932087457217734498320706156564427350079582065129891877, (6, 4) = -.6878860361058951356548892064135523542261590939207278512789580012, (6, 5) = 1.}, order = C_order), Array(1..6, 1..5, {(1, 1) = 0., (1, 2) = 0., (1, 3) = 0., (1, 4) = 0., (1, 5) = 0., (2, 1) = -5.668800000000000000000000000000000000000000000000000000000000000, (2, 2) = 0., (2, 3) = 0., (2, 4) = 0., (2, 5) = 0., (3, 1) = -2.430093356833758185127687948588976454404005541631312071262454980, (3, 2) = -.2063599157089122366435473360234334956951235417816684354325383024, (3, 3) = 0., (3, 4) = 0., (3, 5) = 0., (4, 1) = -.1073529058145262184538714363347072593003701641367435326353413741, (4, 2) = -9.594562251021894319543276722003535660080344141196410019145778438, (4, 3) = -20.47028614809615481077194109199315898088200340959033509022289225, (4, 4) = 0., (4, 5) = 0., (5, 1) = 7.496443313968615025810288671343666721107285253609248916258059945, (5, 2) = -10.24680431464121831828006273262229471844131638878149504437463203, (5, 3) = -33.99990352819906349940544178176705831322109246890624991406278439, (5, 4) = 11.70890893206159543814142227722462662960960061382390320683124096, (5, 5) = 0., (6, 1) = 8.083246795922410929084950629679305621594132916599038753844513444, (6, 2) = -7.981132988062785478298820042190340977772368320604888739105933358, (6, 3) = -31.52159432874372905443421457849868522633084388636090049512840108, (6, 4) = 16.31930543123136189292440273330566432195991444062144703602452206, (6, 5) = -6.058818238834053763270465539455565917330674970761091397715427530}, order = C_order), Array(1..3, 1..5, {(1, 1) = 0., (1, 2) = 0., (1, 3) = 0., (1, 4) = 0., (1, 5) = 0., (2, 1) = 10.12623508344691205709134621774984940971246648780715943495568939, (2, 2) = -7.487995877607633133323634619894111057620617838846132481585712194, (2, 3) = -34.80091861555741139791663438824603908149631772071391291518811903, (2, 4) = -7.992771707568727337179005908592456055163586230675373208664590868, (2, 5) = 1.025137723295620644395805647149506134746459271098695362956446239, (3, 1) = -.6762803392806897770286597359671161843340859823664459332682862303, (3, 2) = 6.087714651678606446194334691957369595295881130569613544529189196, (3, 3) = 16.43084320892463064688438342967802054358092944228249438052416001, (3, 4) = 24.76722511418365125296407852510338208253437292171867382245261245, (3, 5) = -6.594389125716781611244460448890686707490924770268209274199463125}, order = C_order)]), ( 9 ) = ([Array(1..2, {(1) = .1000000000000000000000000000000000000000000000000000000000000000, (2) = .1000000000000000000000000000000000000000000000000000000000000000}, order = C_order), Array(1..2, {(1) = 0., (2) = 0.}, order = C_order), Array(1..2, {(1) = 0., (2) = 0.}, order = C_order), Array(1..2, {(1) = 0., (2) = 0.}, order = C_order), Array(1..2, {(1) = 0., (2) = 0.}, order = C_order), Array(1..2, 1..2, {(1, 1) = 0., (1, 2) = 0., (2, 1) = 0., (2, 2) = 0.}, order = C_order), Array(1..2, 1..2, {(1, 1) = 0., (1, 2) = 0., (2, 1) = 0., (2, 2) = 0.}, order = C_order), Array(1..2, 1..6, {(1, 1) = 0., (1, 2) = 0., (1, 3) = 0., (1, 4) = 0., (1, 5) = 0., (1, 6) = 0., (2, 1) = 0., (2, 2) = 0., (2, 3) = 0., (2, 4) = 0., (2, 5) = 0., (2, 6) = 0.}, order = C_order), Array(1..2, {(1) = 0, (2) = 0}, datatype = integer[8]), Array(1..2, {(1) = 147054706898.100, (2) = 0.}, order = C_order), Array(1..2, {(1) = 0., (2) = 0.}, order = C_order), Array(1..2, {(1) = 0., (2) = 0.}, order = C_order), Array(1..2, {(1) = 0., (2) = 0.}, order = C_order)]), ( 8 ) = ([Array(1..2, {(1) = 0., (2) = 0.}, order = C_order), Array(1..2, {(1) = 0., (2) = 0.}, order = C_order), Array(1..2, {(1) = 0., (2) = 765063.93598266830142830178632448711598260465229563475092800565}, order = C_order)]), ( 11 ) = (Array(1..6, 0..2, {(1, 1) = 0., (1, 2) = 0., (2, 0) = 0., (2, 1) = 0., (2, 2) = 0., (3, 0) = 0., (3, 1) = 0., (3, 2) = 0., (4, 0) = 0., (4, 1) = 0., (4, 2) = 0., (5, 0) = 0., (5, 1) = 0., (5, 2) = 0., (6, 0) = 0., (6, 1) = 0., (6, 2) = 0.}, order = C_order)), ( 10 ) = ([proc (N, X, Y, YP) option `[Y[1] = r(t), Y[2] = diff(r(t),t)]`; YP[2] := 148119034496962719477804289013145600000000.0000000000000000000000/Y[1]^3-990693056236769280000000000000/Y[1]^2; YP[1] := Y[2]; 0 end proc, -1, 0, 0, 0, 0, 0, 0]), ( 13 ) = (), ( 12 ) = (), ( 15 ) = ("rkf45"), ( 14 ) = ([0, 0]), ( 18 ) = ([]), ( 19 ) = (0), ( 16 ) = ([0, 0, 0, []]), ( 17 ) = ([proc (N, X, Y, YP) option `[Y[1] = r(t), Y[2] = diff(r(t),t)]`; YP[2] := 148119034496962719477804289013145600000000.0000000000000000000000/Y[1]^3-990693056236769280000000000000/Y[1]^2; YP[1] := Y[2]; 0 end proc, -1, 0, 0, 0, 0, 0, 0]), ( 20 ) = ([])  ] ))  ] ); _y0 := Array(0..2, {(1) = 0., (2) = 147054706898.100}); _vmap := array( 1 .. 2, [( 1 ) = (1), ( 2 ) = (2)  ] ); _x0 := _dtbl[1][5][5]; _n := _dtbl[1][4][1]; _ne := _dtbl[1][4][3]; _nd := _dtbl[1][4][4]; _nv := _dtbl[1][4][16]; if not type(_xout, 'numeric') then if member(_xout, ["start", "left", "right"]) then if _Env_smart_dsolve_numeric = true or _dtbl[1][4][10] = 1 then if _xout = "left" then if type(_dtbl[2], 'table') then return _dtbl[2][5][1] end if elif _xout = "right" then if type(_dtbl[3], 'table') then return _dtbl[3][5][1] end if end if end if; return _dtbl[1][5][5] elif _xout = "method" then return _dtbl[1][15] elif _xout = "storage" then return evalb(_dtbl[1][4][10] = 1) elif _xout = "leftdata" then if not type(_dtbl[2], 'array') then return NULL else return eval(_dtbl[2]) end if elif _xout = "rightdata" then if not type(_dtbl[3], 'array') then return NULL else return eval(_dtbl[3]) end if elif _xout = "enginedata" then return eval(_dtbl[1]) elif _xout = "enginereset" then _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); return NULL elif _xout = "initial" then return procname(_y0[0]) elif _xout = "laxtol" then return _dtbl[`if`(member(_dtbl[4], {2, 3}), _dtbl[4], 1)][5][18] elif _xout = "numfun" then return `if`(member(_dtbl[4], {2, 3}), _dtbl[_dtbl[4]][4][18], 0) elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return procname(_y0[0]), [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "last" then if _dtbl[4] <> 2 and _dtbl[4] <> 3 or _x0-_dtbl[_dtbl[4]][5][1] = 0. then error "no information is available on last computed point" else _xout := _dtbl[_dtbl[4]][5][1] end if elif _xout = "function" then if _dtbl[1][4][33]-2. = 0 then return eval(_dtbl[1][10], 1) else return eval(_dtbl[1][10][1], 1) end if elif _xout = "map" then return copy(_vmap) elif type(_xin, `=`) and type(rhs(_xin), 'list') and member(lhs(_xin), {"initial", "parameters", "initial_and_parameters"}) then _ini, _par := [], []; if lhs(_xin) = "initial" then _ini := rhs(_xin) elif lhs(_xin) = "parameters" then _par := rhs(_xin) elif select(type, rhs(_xin), `=`) <> [] then _par, _ini := selectremove(type, rhs(_xin), `=`) elif nops(rhs(_xin)) < nops(_pars)+1 then error "insufficient data for specification of initial and parameters" else _par := rhs(_xin)[-nops(_pars) .. -1]; _ini := rhs(_xin)[1 .. -nops(_pars)-1] end if; _xout := lhs(_xout); if _par <> [] then `dsolve/numeric/process_parameters`(_n, _pars, _par, _y0) end if; if _ini <> [] then `dsolve/numeric/process_initial`(_n-_ne, _ini, _y0, _pars, _vmap) end if; `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars); if _Env_smart_dsolve_numeric = true and type(_y0[0], 'numeric') and _dtbl[1][4][10] <> 1 then procname("right") := _y0[0]; procname("left") := _y0[0] end if; if _xout = "initial" then return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)] elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] else return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)], [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] end if elif _xin = "eventstop" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then return 0 end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 10 and 10 <= _dtbl[5-_i][4][9] then _i := 5-_i; _dtbl[4] := _i; _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) elif 10 <= _dtbl[_i][4][9] then _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) else return 0 end if elif _xin = "eventstatus" then if _nv = 0 then error "this solution has no events" end if; _i := [selectremove(proc (a) options operator, arrow; _dtbl[1][3][1][a, 7] = 1 end proc, {seq(_j, _j = 1 .. round(_dtbl[1][3][1][_nv+1, 1]))})]; return ':-enabled' = _i[1], ':-disabled' = _i[2] elif _xin = "eventclear" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then error "no events to clear" end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 10 and 10 < _dtbl[5-_i][4][9] then _dtbl[4] := 5-_i; _i := 5-_i end if; if _dtbl[_i][4][9] < 10 then error "no events to clear" elif _nv < _dtbl[_i][4][9]-10 then error "event error condition cannot be cleared" else _j := _dtbl[_i][4][9]-10; if irem(round(_dtbl[_i][3][1][_j, 4]), 2) = 1 then error "retriggerable events cannot be cleared" end if; _j := round(_dtbl[_i][3][1][_j, 1]); for _k to _nv do if _dtbl[_i][3][1][_k, 1] = _j then if _dtbl[_i][3][1][_k, 2] = 3 then error "range events cannot be cleared" end if; _dtbl[_i][3][1][_k, 8] := _dtbl[_i][3][1][_nv+1, 8] end if end do; _dtbl[_i][4][17] := 0; _dtbl[_i][4][9] := 0; if _dtbl[1][4][10] = 1 then if _i = 2 then try procname(procname("left")) catch:  end try else try procname(procname("right")) catch:  end try end if end if end if; return  elif type(_xin, `=`) and member(lhs(_xin), {"eventdisable", "eventenable"}) then if _nv = 0 then error "this solution has no events" end if; if type(rhs(_xin), {('list')('posint'), ('set')('posint')}) then _i := {op(rhs(_xin))} elif type(rhs(_xin), 'posint') then _i := {rhs(_xin)} else error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; if select(proc (a) options operator, arrow; _nv < a end proc, _i) <> {} then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _k := {}; for _j to _nv do if member(round(_dtbl[1][3][1][_j, 1]), _i) then _k := `union`(_k, {_j}) end if end do; _i := _k; if lhs(_xin) = "eventdisable" then _dtbl[4] := 0; _j := [evalb(assigned(_dtbl[2]) and member(_dtbl[2][4][17], _i)), evalb(assigned(_dtbl[3]) and member(_dtbl[3][4][17], _i))]; for _k in _i do _dtbl[1][3][1][_k, 7] := 0; if assigned(_dtbl[2]) then _dtbl[2][3][1][_k, 7] := 0 end if; if assigned(_dtbl[3]) then _dtbl[3][3][1][_k, 7] := 0 end if end do; if _j[1] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[2][3][4][_k, 1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to defined init `, _dtbl[2][3][4][_k, 1]); _dtbl[2][3][1][_k, 8] := _dtbl[2][3][4][_k, 1] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to rate hysteresis init `, _dtbl[2][5][24]); _dtbl[2][3][1][_k, 8] := _dtbl[2][5][24] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to initial init `, _x0); _dtbl[2][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to fireinitial init `, _x0-1); _dtbl[2][3][1][_k, 8] := _x0-1 end if end do; _dtbl[2][4][17] := 0; _dtbl[2][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("left")) end if end if; if _j[2] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[3][3][4][_k, 2], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to defined init `, _dtbl[3][3][4][_k, 2]); _dtbl[3][3][1][_k, 8] := _dtbl[3][3][4][_k, 2] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to rate hysteresis init `, _dtbl[3][5][24]); _dtbl[3][3][1][_k, 8] := _dtbl[3][5][24] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to initial init `, _x0); _dtbl[3][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to fireinitial init `, _x0+1); _dtbl[3][3][1][_k, 8] := _x0+1 end if end do; _dtbl[3][4][17] := 0; _dtbl[3][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("right")) end if end if else for _k in _i do _dtbl[1][3][1][_k, 7] := 1 end do; _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); _dtbl[4] := 0; if _dtbl[1][4][10] = 1 then if _x0 <= procname("right") then try procname(procname("right")) catch:  end try end if; if procname("left") <= _x0 then try procname(procname("left")) catch:  end try end if end if end if; return  elif type(_xin, `=`) and lhs(_xin) = "eventfired" then if not type(rhs(_xin), 'list') then error "'eventfired' must be specified as a list" end if; if _nv = 0 then error "this solution has no events" end if; if _dtbl[4] <> 2 and _dtbl[4] <> 3 then error "'direction' must be set prior to calling/setting 'eventfired'" end if; _i := _dtbl[4]; _val := NULL; if not assigned(_EnvEventRetriggerWarned) then _EnvEventRetriggerWarned := false end if; for _k in rhs(_xin) do if type(_k, 'integer') then _src := _k elif type(_k, 'integer' = 'anything') and type(evalf(rhs(_k)), 'numeric') then _k := lhs(_k) = evalf[max(Digits, 18)](rhs(_k)); _src := lhs(_k) else error "'eventfired' entry is not valid: %1", _k end if; if _src < 1 or round(_dtbl[1][3][1][_nv+1, 1]) < _src then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _src := {seq(`if`(_dtbl[1][3][1][_j, 1]-_src = 0., _j, NULL), _j = 1 .. _nv)}; if nops(_src) <> 1 then error "'eventfired' can only be set/queried for root-finding events and time/interval events" end if; _src := _src[1]; if _dtbl[1][3][1][_src, 2] <> 0. and _dtbl[1][3][1][_src, 2]-2. <> 0. then error "'eventfired' can only be set/queried for root-finding events and time/interval events" elif irem(round(_dtbl[1][3][1][_src, 4]), 2) = 1 then if _EnvEventRetriggerWarned = false then WARNING(`'eventfired' has no effect on events that retrigger`) end if; _EnvEventRetriggerWarned := true end if; if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then _val := _val, undefined elif type(_dtbl[_i][3][4][_src, _i-1], 'undefined') or _i = 2 and _dtbl[2][3][1][_src, 8] < _dtbl[2][3][4][_src, 1] or _i = 3 and _dtbl[3][3][4][_src, 2] < _dtbl[3][3][1][_src, 8] then _val := _val, _dtbl[_i][3][1][_src, 8] else _val := _val, _dtbl[_i][3][4][_src, _i-1] end if; if type(_k, `=`) then if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then error "cannot set event code for a rate hysteresis event" end if; userinfo(3, {'events', 'eventreset'}, `manual set event code `, _src, ` to value `, rhs(_k)); _dtbl[_i][3][1][_src, 8] := rhs(_k); _dtbl[_i][3][4][_src, _i-1] := rhs(_k) end if end do; return [_val] elif type(_xin, `=`) and lhs(_xin) = "direction" then if not member(rhs(_xin), {-1, 1, ':-left', ':-right'}) then error "'direction' must be specified as either '1' or 'right' (positive) or '-1' or 'left' (negative)" end if; _src := `if`(_dtbl[4] = 2, -1, `if`(_dtbl[4] = 3, 1, undefined)); _i := `if`(member(rhs(_xin), {1, ':-right'}), 3, 2); _dtbl[4] := _i; _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if; return _src elif _xin = "eventcount" then if _dtbl[1][3][1] = 0 or _dtbl[4] <> 2 and _dtbl[4] <> 3 then return 0 else return round(_dtbl[_dtbl[4]][3][1][_nv+1, 12]) end if else return "procname" end if end if; if _xout = _x0 then return [_x0, seq(evalf(_dtbl[1][6][_vmap[_i]]), _i = 1 .. _n-_ne)] end if; _i := `if`(_x0 <= _xout, 3, 2); if _xin = "last" and 0 < _dtbl[_i][4][9] and _dtbl[_i][4][9] < 10 then _dat := eval(_dtbl[_i], 2); _j := _dat[4][20]; return [_dat[11][_j, 0], seq(_dat[11][_j, _vmap[_i]], _i = 1 .. _n-_ne-_nd), seq(_dat[8][1][_vmap[_i]], _i = _n-_ne-_nd+1 .. _n-_ne)] end if; if not type(_dtbl[_i], 'array') then _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if end if; if _xin <> "last" then if 0 < 0 then if `dsolve/numeric/checkglobals`(op(_dtbl[1][14]), _pars, _n, _y0) then `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars, _i) end if end if; if _dtbl[1][4][7] = 0 then error "parameters must be initialized before solution can be computed" end if end if; _dat := eval(_dtbl[_i], 2); _dtbl[4] := _i; try _src := `dsolve/numeric/SC/IVPrun`(_dat, _xout) catch: userinfo(2, `dsolve/debug`, print(`Exception in solnproc:`, [lastexception][2 .. -1])); error  end try; if _src = 0 and 10 < _dat[4][9] then _val := _dat[3][1][_nv+1, 8] else _val := _dat[11][_dat[4][20], 0] end if; if _src <> 0 or _dat[4][9] <= 0 then _dtbl[1][5][1] := _xout else _dtbl[1][5][1] := _val end if; if _i = 3 and _val < _xout then Rounding := -infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further right of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further right of %1, maxfun limit exceeded (see <a href='http://www.maplesoft.com/support/help/search.aspx?term=dsolve,maxfun' target='_new'>?dsolve,maxfun</a> for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further right of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further right of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif 10 < _dat[4][9] then if _dat[4][9]-10 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-10 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-10 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further right of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-10, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further right of %1", evalf[8](_val) end if elif _i = 2 and _xout < _val then Rounding := infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further left of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further left of %1, maxfun limit exceeded (see <a href='http://www.maplesoft.com/support/help/search.aspx?term=dsolve,maxfun' target='_new'>?dsolve,maxfun</a> for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further left of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further left of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif 10 < _dat[4][9] then if _dat[4][9]-10 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-10 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-10 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further left of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-10, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further left of %1", evalf[8](_val) end if end if; if _EnvInFsolve = true then _dig := _dat[4][26]; _dat[4][26] := _EnvDSNumericSaveDigits; _Env_dsolve_SC_native := true; if _dat[4][25] = 1 then _i := 1; _dat[4][25] := 2 else _i := _dat[4][25] end if; _val := `dsolve/numeric/SC/IVPval`(_dat, _xout, _src); _dat[4][25] := _i; _dat[4][26] := _dig; [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] else Digits := _dat[4][26]; _val := `dsolve/numeric/SC/IVPval`(eval(_dat, 2), _xout, _src); [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] end if end proc, (2) = Array(0..0, {}), (3) = [t, r(t), diff(r(t), t)], (4) = []}); _vars := _dat[3]; _pars := map(rhs, _dat[4]); _n := nops(_vars)-1; _solnproc := _dat[1]; if not type(_xout, 'numeric') then if member(x_rkf45, ["start", 'start', "method", 'method', "left", 'left', "right", 'right', "leftdata", "rightdata", "enginedata", "eventstop", 'eventstop', "eventclear", 'eventclear', "eventstatus", 'eventstatus', "eventcount", 'eventcount', "laxtol", 'laxtol', "numfun", 'numfun', NULL]) then _res := _solnproc(convert(x_rkf45, 'string')); if 1 < nops([_res]) then return _res elif type(_res, 'array') then return eval(_res, 1) elif _res <> "procname" then return _res end if elif member(x_rkf45, ["last", 'last', "initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(x_rkf45, 'string'); _res := _solnproc(_xout); if _xout = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] end if elif type(_xout, `=`) and member(lhs(_xout), ["initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(lhs(x_rkf45), 'string') = rhs(x_rkf45); if type(rhs(_xout), 'list') then _res := _solnproc(_xout) else error "initial and/or parameter values must be specified in a list" end if; if lhs(_xout) = "initial" then return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] elif lhs(_xout) = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] end if elif type(_xout, `=`) and member(lhs(_xout), ["eventdisable", 'eventdisable', "eventenable", 'eventenable', "eventfired", 'eventfired', "direction", 'direction', NULL]) then return _solnproc(convert(lhs(x_rkf45), 'string') = rhs(x_rkf45)) elif _xout = "solnprocedure" then return eval(_solnproc) elif _xout = "sysvars" then return _vars end if; if procname <> unknown then return ('procname')(x_rkf45) else _ndsol; _ndsol := pointto(_dat[2][0]); return ('_ndsol')(x_rkf45) end if end if; try _res := _solnproc(_xout); [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] catch: error  end try end proc

(2)

sol(182)

[t = 182., r(t) = 152049647052.7850981852952995533786837204381025832707997878034578, diff(r(t), t) = 389930.6771708058608633758773386707775821562141034647310382334420]

(3)

rt := proc (t) options operator, arrow; rhs(op(2, sol(t))) end proc

proc (t) options operator, arrow; rhs(op(2, sol(t))) end proc

(4)

plot(rt/AE, 0 .. 400)

 

``

restart; Digits := 64; assume(r > 0); epsilon := 0.167e-1; AE := 149597870700; mu := 1; v_T := 30290; r_0 := .983*AE; GM := 132712440018*10^9; L_0 := sqrt(GM*mu^2*r_0); d := 86400; M_0 := 10*Heaviside(365.25-t); ode := diff(r(t), t, t)+(-(L_0+M_0*t)^2/(mu^2*r(t)^3)+GM/r(t)^2)*d^2 = 0

diff(diff(r(t), t), t)-7464960000*(10*Heaviside(365.25-t)*t+4417690456401248.026238441367771530784192296996671291816290374426)^2/r(t)^3+990693056236769280000000000000/r(t)^2 = 0

(5)

sol := dsolve({ode, r(0) = r_0, (D(r))(0) = 0}, r(t), numeric);

proc (t) options operator, arrow; rhs(op(3, sol(t))) end proc

(6)

plot({rt/AE}, 0 .. 800)

 

plot({(1/86400)*rpt}, 0 .. 800)

 

(D(rpt))(t)

(D(rpt))(t)

(7)



Download Kepler_Gravitatio.mw

 

Now, don't worry about the upper part ending with the first plot, this was just for startup.

I tried to emulate an external moment of force which acts tangential on a particle of unit mass. After one year = 365,25 days I switch of the force which was done via heaviside function.

I'm wondering about the results. Shouldn't the function r(t) be constant for t>365.25 since D(r)(365.25) and the also the second derivative vanishes. Unfortunately this is not represented by the results of the ode for t>365,25

What problems do occur here and how can I omit them?

BTW: how do I derive the numeric function rt or rpt with respect to t

e.g. D(rpt)(10) does not work

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