Kitonum

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11 years, 362 days

MaplePrimes Activity


These are answers submitted by Kitonum

Your surface is a torus:

restart;
plot3d(eval([x,sqrt(x^2+y^2)*cos(s),sqrt(x^2+y^2)*sin(s)],[x=cos(t),y=2+sin(t)]), t=0..2*Pi, s=0..2*Pi, scaling=constrained, labels=[x,y,z]);

                       

 


 

restart;
r1:=2:  r2:=2*(1-cos(theta)):
A:=plot([r1,r2], theta=0..2*Pi, color=[red,blue], thickness=3, coords=polar):
B:=plot(r1, theta=Pi/2..3*Pi/2, color=green, coords=polar, filled):
C:=plot(r2, theta=-Pi/2..Pi/2, color=green, coords=polar, filled):
plots:-display(A,B,C, scaling=constrained);

 

 


 

Download shade.mw


Here is another shorter way:
 

restart;
r1:=2:  r2:=2*(1-cos(theta)):
plot([r1,r2], theta=0..2*Pi, color=[red,blue], thickness=3, coords=polar, filled, scaling=constrained);

 

 


 

Download shade1.mw


Here is a more automatic plotting for the intersection:

restart;
r1:=2:  r2:=2*(1-cos(theta)):
plot(min(r1,r2), theta=0..2*Pi, color=green, coords=polar, filled, scaling=constrained);

Edit.

The following approach is sometimes useful:

restart;
eq := x^2+floor(x)-10:
Student:-Calculus1:-Roots(eq);
identify(%);

    

 

The animation you see here is saved in GIF format. Each polygon from the list remains on the display for exactly 1 second (10 frames per second by default):

restart;
OneFrame:=proc(n)
local Circle, Polygon, Text, S;
uses plottools, plots;
Circle:=plot([cos(t),sin(t),t=0..2*Pi], color=grey, thickness=2);
Polygon:=polygon([seq([cos(2*Pi*k/n),sin(2*Pi*k/n)], k=1..n)], color=yellow);
S:=evalf(n*sin(2*Pi/n)/2);
Text:=textplot([[-0.7,-1.1,"Area of Circle = 3.14"],[0,-1.1,typeset("n","=",n)],[0.7,-1.1,typeset("Area of Polygon","=",evalf[3](S))]], font=[times,16]);

display(Text,Circle,Polygon, scaling=constrained, axes=none, title="Archimedes Approximation for Pi \n", titlefont=[times,bold,18]);
end proc:

plots:-display(seq(OneFrame(n)$10, n=[3,6,9,12,15,18,21,24,27,30,60]), insequence, size=[600,600]); 

                       

 


 

restart;
sys_ode := diff(F0(zeta), zeta, zeta)-b^2*F0(zeta)+G0(zeta)^2 = 0, diff(G0(zeta), zeta, zeta)-b^2*G0(zeta) = 0, 2*F0(zeta)+diff(H0(zeta), zeta) = 0;
ics := F0(0) = 0, G0(0) = 1, H0(0) = 0, F0(infinity) = 0, G0(infinity) = 0;

sol:=dsolve([sys_ode,ics]);
sol1:=eval(sol,b=1):
plot([eval(F0(zeta),sol1),eval(G0(zeta),sol1),eval(H0(zeta),sol1)], zeta=0..10, color=[red,blue,green]);

diff(diff(F0(zeta), zeta), zeta)-b^2*F0(zeta)+G0(zeta)^2 = 0, diff(diff(G0(zeta), zeta), zeta)-b^2*G0(zeta) = 0, 2*F0(zeta)+diff(H0(zeta), zeta) = 0

 

F0(0) = 0, G0(0) = 1, H0(0) = 0, F0(infinity) = 0, G0(infinity) = 0

 

{F0(zeta) = limit((1/2)*(-(1/3)*exp(b*zeta)*((exp(b*_a))^3*(exp(-b*_a))^3-9*(exp(b*_a))^2*(exp(-b*_a))^2+2*exp(-b*_a)*(exp(b*_a))^2+12*(exp(-b*_a))^2*exp(b*_a)+2*(exp(-b*_a))^3-9*exp(-b*_a)*exp(b*_a)+1)/(b*((exp(b*_a))^3-3*exp(-b*_a)*(exp(b*_a))^2+3*(exp(-b*_a))^2*exp(b*_a)-(exp(-b*_a))^3))+(1/3)*exp(-b*zeta)*((exp(b*_a))^3*(exp(-b*_a))^3-9*(exp(b*_a))^2*(exp(-b*_a))^2+2*(exp(b*_a))^3+12*exp(-b*_a)*(exp(b*_a))^2+2*(exp(-b*_a))^2*exp(b*_a)-9*exp(-b*_a)*exp(b*_a)+1)/(((exp(b*_a))^2-2*exp(-b*_a)*exp(b*_a)+(exp(-b*_a))^2)*b*(exp(b*_a)-exp(-b*_a)))-(-(1/3)*(exp(b*_a))^2/((exp(b*_a)-exp(-b*_a))^2*b*(exp(b*zeta))^3)+(exp(-b*_a))^2*exp(b*zeta)/((exp(b*_a)-exp(-b*_a))^2*b)+2*exp(-b*_a)*exp(b*_a)/((exp(b*_a)-exp(-b*_a))^2*b*exp(b*zeta)))*exp(b*zeta)+((1/3)*(exp(-b*_a))^2*(exp(b*zeta))^3/((exp(b*_a)-exp(-b*_a))^2*b)-(exp(b*_a))^2/((exp(b*_a)-exp(-b*_a))^2*b*exp(b*zeta))-2*exp(-b*_a)*exp(b*_a)*exp(b*zeta)/((exp(b*_a)-exp(-b*_a))^2*b))*exp(-b*zeta))/b, _a = infinity), G0(zeta) = limit(-exp(-b*_a)*exp(b*zeta)/(exp(b*_a)-exp(-b*_a))+exp(b*_a)*exp(-b*zeta)/(exp(b*_a)-exp(-b*_a)), _a = infinity), H0(zeta) = limit(-(1/3)*(-12*exp(-b*_a)*exp(b*_a)*zeta/(exp(b*_a)-exp(-b*_a))^2+(exp(b*_a))^2/((exp(b*_a)-exp(-b*_a))^2*(exp(b*zeta))^2*b)-(exp(-b*_a))^2*(exp(b*zeta))^2/((exp(b*_a)-exp(-b*_a))^2*b)-((exp(b*_a))^3*(exp(-b*_a))^3-9*(exp(b*_a))^2*(exp(-b*_a))^2+2*(exp(b*_a))^3+12*exp(-b*_a)*(exp(b*_a))^2+2*(exp(-b*_a))^2*exp(b*_a)-9*exp(-b*_a)*exp(b*_a)+1)/(((exp(b*_a))^2-2*exp(-b*_a)*exp(b*_a)+(exp(-b*_a))^2)*b*(exp(b*_a)-exp(-b*_a))*exp(b*zeta))-exp(b*zeta)*((exp(b*_a))^3*(exp(-b*_a))^3-9*(exp(b*_a))^2*(exp(-b*_a))^2+2*exp(-b*_a)*(exp(b*_a))^2+12*(exp(-b*_a))^2*exp(b*_a)+2*(exp(-b*_a))^3-9*exp(-b*_a)*exp(b*_a)+1)/(b*((exp(b*_a))^3-3*exp(-b*_a)*(exp(b*_a))^2+3*(exp(-b*_a))^2*exp(b*_a)-(exp(-b*_a))^3)))/b^2-(1/3)*(2*(exp(b*_a))^3*(exp(-b*_a))^3-18*(exp(b*_a))^2*(exp(-b*_a))^2+(exp(b*_a))^3+15*exp(-b*_a)*(exp(b*_a))^2+15*(exp(-b*_a))^2*exp(b*_a)+(exp(-b*_a))^3-18*exp(-b*_a)*exp(b*_a)+2)/(b^3*((exp(b*_a))^3-3*exp(-b*_a)*(exp(b*_a))^2+3*(exp(-b*_a))^2*exp(b*_a)-(exp(-b*_a))^3)), _a = infinity)}

 

 

 


 

Download diffeq.mw

Maple has the command  Student:-LinearAlgebra:-LinearSolveTutor  that solves systems of linear equations step by step, but unfortunately only if the matrix of the system is no more than 5 by 5. Below is a step-by-step solution using the Jordan Gauss method using my program  JordanGausse (all comments are in Russian):

system.mw

reduce the range along the vertical axis (y-range) while maintaining  x-range. Also use a list rather than a set when specifying functions so that all options match the corresponding function:

restart;
plot([0, 2*x^2, 2*x^2 - 2*x^3 + 8/3*x^4 - 4*x^5], x = -10 .. 10, y=-1000..1000, color = ["DarkGreen", "CornflowerBlue", "Burgundy"], axes=box);

                       

 

Here is another way by the using the  applyrule  command:

restart;
expr:=1/exp(z)*arcsinh(x*exp(C[1]))+x*sin(exp(x))+3*exp(C[1]*y)*sqrt(sin(exp(3*h)));
applyrule(exp(t::anything)=Z, expr);

 

I think that there are infinitely many ellipses inscribed in a given triangle. But among them there will be only one that touches the sides of the triangle in their midpoints. It is called the Steiner ellipse. See  https://en.wikipedia.org/wiki/Steiner_inellipse

Here is an example of plotting this ellipse:
 

restart;
A:=<0,0>: B:=<5,6>: C:=<4,0>:
S:=1/3*(A+B+C);
AS:=S-A: SC:=C-S: AB:=B-A:
XY:=AS+1/2*SC*cos(t)+1/2/sqrt(3)*AB*sin(t);
ABC:=plottools:-curve(convert~([A,B,C,A],list), color=blue):
P:=plots:-pointplot(convert([(A+B)/2,(B+C)/2,(A+C)/2],list), symbol=solidcircle, color=red, symbolsize=12):
plots:-display(plot([XY[1],XY[2],t=0..2*Pi], color=red), ABC, P, scaling=constrained);

Vector(2, {(1) = 3, (2) = 2})

 

Vector[column](%id = 18446746141167283846)

 

 

 


As for orthoptic circle for an ellipse see  https://en.wikipedia.org/wiki/Orthoptic_(geometry)#:~:text=Generalizations%3A,fixed%20angle%20(see%20below).&text=Thales'%20theorem%20on%20a%20chord,two%20points%20P%20and%20Q.

 

Download Steiner_ellipse.mw

For a rational function  f(x)=P(x)/Q(x)  to have a horizontal asymptote  y=y0  and  y0<>0 , it is necessary and sufficient that the polynomials  P(x)  and  Q(x)  have the same degree. In this case, the asymptote will be the same at + infinity and -infinity. Thus, a rational function cannot have two different horizontal asymptotes. Therefore, your conditions may be only partially implemented.

Here is an example:

restart;
f:=x->0.7*x^2/(x-0.001)/(x+0.001);
plot([f(x),0.7,[-0.001,t,t=-1..2],[0.001,t,t=-1..2]], x=-0.005..0.005,-1..2, linestyle=[1,3$3], color=[red,black$3], thickness=[2,1$3], discont, size=[500,500]);

proc (x) options operator, arrow; .7*(x^2)/((x-0.1e-2)*(x+0.1e-2)) end proc

 

 

 


 

Download rational_function.mw


 

restart;
c1:=3.2: c2:=3.3: c3:=3.4: R:=-10: A:=1.6:

sol:=dsolve({diff(f(x),x$4) - c1*diff(g(x),x$2) + R*(diff(f(x),x)* diff(f(x),x$2) - f(x)*diff(f(x),x$3))=0,diff(g(x),x$2)+c2*(diff(f(x),x$2)-2*g(x))-c3*(f(x)*diff(g(x),x)-diff(f(x),x)*g(x))=0,D(f)(-1)=0, D(f)(1)=0,f(-1)=1-A,f(1) =1,g(-1)=0,g(1)=0}, numeric);

proc (x_bvp) local res, data, solnproc, _ndsol, outpoint, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then outpoint := evalf[_EnvDSNumericSaveDigits](x_bvp) else outpoint := evalf(x_bvp) 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) = -1.0, (2) = -.9379808670115692, (3) = -.8737879401746669, (4) = -.8071752365321642, (5) = -.737846554366134, (6) = -.6647730167481146, (7) = -.5846962906002146, (8) = -.4948419722955886, (9) = -.39095850902064355, (10) = -.2727817089533898, (11) = -.13560606119241353, (12) = 0.15609899327280935e-1, (13) = .17138179998337502, (14) = .32004065269496373, (15) = .4574141854843416, (16) = .5796079126672031, (17) = .6886347209415522, (18) = .7846266562404318, (19) = .8653413901795519, (20) = .9361519290003208, (21) = 1.0}, datatype = float[8], order = C_order); Y := Matrix(21, 6, {(1, 1) = -.6, (1, 2) = .0, (1, 3) = 1.933574172895688, (1, 4) = 3.2935327906510654, (1, 5) = .0, (1, 6) = 2.989334789810824, (2, 1) = -.5961626701578583, (2, 2) = .12546644498862447, (2, 3) = 2.1000534596203093, (2, 4) = 2.090992178215719, (2, 5) = .16269807206994533, (2, 6) = 2.277382088918899, (3, 1) = -.5837025183977008, (3, 2) = .26377828746180204, (3, 3) = 2.196917150370234, (3, 4) = .9427342310502282, (3, 5) = .28835968963429326, (3, 6) = 1.6553960068739786, (4, 1) = -.5612247767453585, (4, 2) = .4113861423699064, (4, 3) = 2.222781700020577, (4, 4) = -.14928215645654438, (4, 5) = .3799990966227525, (4, 6) = 1.1116606884000693, (5, 1) = -.527385077285588, (5, 2) = .5642845336873072, (5, 3) = 2.1761822579858228, (5, 4) = -1.1757589520057652, (5, 5) = .44011284779599846, (5, 6) = .6365605592019381, (6, 1) = -.4804332889112951, (6, 2) = .7192893161754781, (6, 3) = 2.0546693921413266, (6, 4) = -2.1263980985175523, (6, 5) = .47090928849845315, (6, 6) = .21946829224336692, (7, 1) = -.4164491770135664, (7, 2) = .8760236754661979, (7, 3) = 1.848339140308811, (7, 4) = -2.99486566803203, (7, 5) = .47299107467294255, (7, 6) = -.1539531348969532, (8, 1) = -.33065976147589365, (8, 2) = 1.0289404469293186, (8, 3) = 1.5443705688574503, (8, 4) = -3.7256130438968027, (8, 5) = .4437076719719681, (8, 6) = -.48300810682801176, (9, 1) = -.21616116481500305, (9, 2) = 1.1682121253580369, (9, 3) = 1.1284035858221788, (9, 4) = -4.218325172351393, (9, 5) = .37825714156206786, (9, 6) = -.7596586345023679, (10, 1) = -0.714031798043051e-1, (10, 2) = 1.271595578001645, (10, 3) = .6192209205456293, (10, 4) = -4.322174364759075, (10, 5) = .27571558419417747, (10, 6) = -.9562772986583437, (11, 1) = .10702098920593602, (11, 2) = 1.3166936422936484, (11, 3) = 0.4684849708352476e-1, (11, 4) = -3.95288179345142, (11, 5) = .1365175395260044, (11, 6) = -1.0518068653915225, (12, 1) = .30448004138140244, (12, 2) = 1.28117401384149, (12, 3) = -.49827772326714526, (12, 4) = -3.229903767975225, (12, 5) = -0.22160310947253917e-1, (12, 6) = -1.0272952971400855, (13, 1) = .4960968363639633, (13, 2) = 1.1675953799259322, (13, 3) = -.9395998350138852, (13, 4) = -2.4462833634708323, (13, 5) = -.17315806804772424, (13, 6) = -.8958170476810169, (14, 1) = .6580438917122471, (14, 2) = 1.0034397433244657, (14, 3) = -1.252120344541368, (14, 4) = -1.7685386814686117, (14, 5) = -.29188336317567026, (14, 6) = -.6884905412860394, (15, 1) = .7833760386113022, (15, 2) = .816634476498511, (15, 3) = -1.453745979067012, (15, 4) = -1.1644466870181112, (15, 5) = -.3690488686597065, (15, 6) = -.4212123297455981, (16, 1) = .8719989299520304, (16, 2) = .6317023102203725, (16, 3) = -1.5613437907241403, (16, 4) = -.5851389666137105, (16, 5) = -.4018282187140272, (16, 6) = -0.9897957049092156e-1, (17, 1) = .9314960462959884, (17, 2) = .4591335944284822, (17, 3) = -1.5933972273271257, (17, 4) = 0.14454651785200952e-1, (17, 5) = -.3923723698236356, (17, 6) = .293052962053937, (18, 1) = .968252273554313, (18, 2) = .3071776561395629, (18, 3) = -1.5624266472437789, (18, 4) = .6523766550656892, (18, 5) = -.3426949154605619, (18, 6) = .7669836164313891, (19, 1) = .9880272727551623, (19, 2) = .18387101334265538, (19, 3) = -1.4840854206694534, (19, 4) = 1.3115956702917055, (19, 5) = -.26003390463049686, (19, 6) = 1.3075390762410004, (20, 1) = .9974140861984128, (20, 2) = 0.8263515866568337e-1, (20, 3) = -1.3668583614521281, (20, 4) = 2.0241991272292283, (20, 5) = -.14619002644293652, (20, 6) = 1.9363934917444479, (21, 1) = 1.0, (21, 2) = .0, (21, 3) = -1.213219755822394, (21, 4) = 2.8158116176608328, (21, 5) = .0, (21, 6) = 2.6741867611734547}, datatype = float[8], order = C_order); YP := Matrix(21, 6, {(1, 1) = .0, (1, 2) = 1.933574172895688, (1, 3) = 3.2935327906510654, (1, 4) = -20.171724029757133, (1, 5) = 2.989334789810824, (1, 6) = -12.479037741769849, (2, 1) = .12546644498862447, (2, 2) = 2.1000534596203093, (2, 3) = 2.090992178215719, (2, 4) = -18.633568323982768, (2, 5) = 2.277382088918899, (2, 6) = -10.54192048283493, (3, 1) = .26377828746180204, (3, 2) = 2.196917150370234, (3, 3) = .9427342310502282, (3, 4) = -17.151996232891765, (3, 5) = 1.6553960068739786, (3, 6) = -8.890546911728242, (4, 1) = .4113861423699064, (4, 2) = 2.222781700020577, (4, 3) = -.14928215645654438, (4, 4) = -15.629356971128114, (4, 5) = 1.1116606884000693, (4, 6) = -7.479926378391591, (5, 1) = .5642845336873072, (5, 2) = 2.1761822579858228, (5, 3) = -1.1757589520057652, (5, 4) = -13.960806821209731, (5, 5) = .6365605592019381, (5, 6) = -6.262465459419724, (6, 1) = .7192893161754781, (6, 2) = 2.0546693921413266, (6, 3) = -2.1263980985175523, (6, 4) = -12.021071149501477, (6, 5) = .21946829224336692, (6, 6) = -5.182551328077048, (7, 1) = .8760236754661979, (7, 2) = 1.848339140308811, (7, 3) = -2.99486566803203, (7, 4) = -9.619681210014855, (7, 5) = -.1539531348969532, (7, 6) = -4.168586329639746, (8, 1) = 1.0289404469293186, (8, 2) = 1.5443705688574503, (8, 3) = -3.7256130438968027, (8, 4) = -6.595488129126174, (8, 5) = -.48300810682801176, (8, 6) = -3.1771994869090414, (9, 1) = 1.1682121253580369, (9, 2) = 1.1284035858221788, (9, 3) = -4.218325172351393, (9, 4) = -2.8844979698863806, (9, 5) = -.7596586345023679, (9, 6) = -2.171332704436469, (10, 1) = 1.271595578001645, (10, 2) = .6192209205456293, (10, 3) = -4.322174364759075, (10, 4) = 1.000341969909436, (10, 5) = -.9562772986583437, (10, 6) = -1.1835856064641754, (11, 1) = 1.3166936422936484, (11, 2) = 0.4684849708352476e-1, (11, 3) = -3.95288179345142, (11, 4) = 4.0553836852185015, (11, 5) = -1.0518068653915225, (11, 6) = -.2474627171510777, (12, 1) = 1.28117401384149, (12, 2) = -.49827772326714526, (12, 3) = -3.229903767975225, (12, 4) = 5.150125875894489, (12, 5) = -1.0272952971400855, (12, 6) = .531099454326164, (13, 1) = 1.1675953799259322, (13, 2) = -.9395998350138852, (13, 3) = -2.4462833634708323, (13, 4) = 4.794786111283175, (13, 5) = -.8958170476810169, (13, 6) = 1.1342425000063217, (14, 1) = 1.0034397433244657, (14, 2) = -1.252120344541368, (14, 3) = -1.7685386814686117, (14, 4) = 4.388656248198329, (14, 5) = -.6884905412860394, (14, 6) = 1.660990204251605, (15, 1) = .816634476498511, (15, 2) = -1.453745979067012, (15, 3) = -1.1644466870181112, (15, 4) = 4.496395866650609, (15, 5) = -.4212123297455981, (15, 6) = 2.2644345012248825, (16, 1) = .6317023102203725, (16, 2) = -1.5613437907241403, (16, 3) = -.5851389666137105, (16, 4) = 5.063219173481798, (16, 5) = -0.9897957049092156e-1, (16, 6) = 3.069955763238939, (17, 1) = .4591335944284822, (17, 2) = -1.5933972273271257, (17, 3) = 0.14454651785200952e-1, (17, 4) = 6.018947447442345, (17, 5) = .293052962053937, (17, 6) = 4.209191850211921, (18, 1) = .3071776561395629, (18, 2) = -1.5624266472437789, (18, 3) = .6523766550656892, (18, 4) = 7.370603573695792, (18, 5) = .7669836164313891, (18, 6) = 5.777087788253219, (19, 1) = .18387101334265538, (19, 2) = -1.4840854206694534, (19, 3) = 1.3115956702917055, (19, 4) = 9.06820312378945, (19, 5) = 1.3075390762410004, (19, 6) = 7.73622779887591, (20, 1) = 0.8263515866568337e-1, (20, 2) = -1.3668583614521281, (20, 3) = 2.0241991272292283, (20, 4) = 11.1722545376268, (20, 5) = 1.9363934917444479, (20, 6) = 10.153564794074244, (21, 1) = .0, (21, 2) = -1.213219755822394, (21, 3) = 2.8158116176608328, (21, 4) = 13.748636406443344, (21, 5) = 2.6741867611734547, (21, 6) = 13.095860182203648}, 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) = -1.0, (2) = -.9379808670115692, (3) = -.8737879401746669, (4) = -.8071752365321642, (5) = -.737846554366134, (6) = -.6647730167481146, (7) = -.5846962906002146, (8) = -.4948419722955886, (9) = -.39095850902064355, (10) = -.2727817089533898, (11) = -.13560606119241353, (12) = 0.15609899327280935e-1, (13) = .17138179998337502, (14) = .32004065269496373, (15) = .4574141854843416, (16) = .5796079126672031, (17) = .6886347209415522, (18) = .7846266562404318, (19) = .8653413901795519, (20) = .9361519290003208, (21) = 1.0}, datatype = float[8], order = C_order); Y := Matrix(21, 6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = 0.20045925470685708e-9, (1, 4) = 0.8277557666289159e-10, (1, 5) = .0, (1, 6) = 0.15114560749105982e-9, (2, 1) = 0.5158879880588224e-12, (2, 2) = 0.14215271091154637e-10, (2, 3) = 0.2004661576674297e-9, (2, 4) = -0.8887372862376951e-10, (2, 5) = 0.757724753264518e-11, (2, 6) = 0.9125228675118423e-10, (3, 1) = 0.2128154366806889e-11, (3, 2) = 0.287530210876931e-10, (3, 3) = 0.18876173516903876e-9, (3, 4) = -0.2540075546656048e-9, (3, 5) = 0.11625606776263755e-10, (3, 6) = 0.40381030163406105e-10, (4, 1) = 0.4963269255937875e-11, (4, 2) = 0.4285152453796549e-10, (4, 3) = 0.16493408755446793e-9, (4, 4) = -0.4053397535086608e-9, (4, 5) = 0.12333029015770294e-10, (4, 6) = -0.1791442382917446e-11, (5, 1) = 0.9116047066444794e-11, (5, 2) = 0.5557925799045456e-10, (5, 3) = 0.12882025439142448e-9, (5, 4) = -0.5332966583436343e-9, (5, 5) = 0.9864457687322937e-11, (5, 6) = -0.35285712141483545e-10, (6, 1) = 0.1470916327690705e-10, (6, 2) = 0.6582207372903393e-10, (6, 3) = 0.795247776128351e-10, (6, 4) = -0.6294952828563126e-9, (6, 5) = 0.4338808545975659e-11, (6, 6) = -0.5975801460942937e-10, (7, 1) = 0.22151231492004536e-10, (7, 2) = 0.7154281326591275e-10, (7, 3) = 0.10655411424085484e-10, (7, 4) = -0.6940013965057971e-9, (7, 5) = -0.449943838899647e-11, (7, 6) = -0.7227437435541223e-10, (8, 1) = 0.3143780948875737e-10, (8, 2) = 0.6535678153294401e-10, (8, 3) = -0.949895573966807e-10, (8, 4) = -0.7449037210337756e-9, (8, 5) = -0.15971470006080478e-10, (8, 6) = -0.5453969567071403e-10, (9, 1) = 0.4092623495751743e-10, (9, 2) = 0.26536027992495592e-10, (9, 3) = -0.2429237937554614e-9, (9, 4) = -0.7138209108619224e-9, (9, 5) = -0.23377481324379515e-10, (9, 6) = 0.5276528968153014e-10, (10, 1) = 0.4415857324275836e-10, (10, 2) = -0.4818863075783663e-10, (10, 3) = -0.19283214523517545e-9, (10, 4) = -0.27860357123918643e-9, (10, 5) = -0.70822957672681055e-11, (10, 6) = 0.25390799175668336e-9, (11, 1) = 0.15523612061727348e-10, (11, 2) = -0.1467030560619035e-10, (11, 3) = 0.23143389392294444e-9, (11, 4) = -0.33032689167286705e-9, (11, 5) = 0.8201024797696422e-10, (11, 6) = 0.8621989339731842e-10, (12, 1) = -0.3641770562042089e-10, (12, 2) = 0.3141601889820978e-9, (12, 3) = -0.13375324270905143e-8, (12, 4) = 0.6162751129968969e-8, (12, 5) = 0.15360239425929725e-9, (12, 6) = -0.8105870771949846e-9, (13, 1) = 0.595678701625678e-10, (13, 2) = -0.13846885765846146e-9, (13, 3) = -0.7955697858200915e-10, (13, 4) = 0.27488667169234544e-8, (13, 5) = -0.16402585258079348e-9, (13, 6) = 0.5119846575641247e-9, (14, 1) = 0.40823635446837066e-10, (14, 2) = -0.2699499524866616e-9, (14, 3) = 0.14760109918441855e-8, (14, 4) = -0.98068495484932e-8, (14, 5) = -0.134851729771407e-9, (14, 6) = 0.6865689447109253e-9, (15, 1) = -0.17210360954127904e-11, (15, 2) = -0.26161657275172888e-10, (15, 3) = -0.16332969772296796e-10, (15, 4) = -0.33046127679841334e-9, (15, 5) = -0.39546372424047783e-10, (15, 6) = -0.14506510660938657e-9, (16, 1) = -0.14336918408313271e-11, (16, 2) = -0.20628752921859975e-10, (16, 3) = -0.725259632009807e-10, (16, 4) = 0.32197505437370873e-9, (16, 5) = -0.4392858251618543e-10, (16, 6) = -0.22314853003345976e-9, (17, 1) = 0.6510741250787862e-12, (17, 2) = -0.28517055679764822e-10, (17, 3) = 0.6972504039318902e-10, (17, 4) = -0.6715203795248354e-9, (17, 5) = -0.3285694273905973e-10, (17, 6) = -0.15865548298597958e-9, (18, 1) = 0.682305383929572e-12, (18, 2) = -0.18654020318015608e-10, (18, 3) = 0.8173214248380389e-10, (18, 4) = -0.5386766276504061e-9, (18, 5) = -0.11214022447994526e-10, (18, 6) = -0.9616707545852851e-10, (19, 1) = 0.1454058879376567e-12, (19, 2) = -0.10081470952846923e-10, (19, 3) = 0.609174267391267e-10, (19, 4) = -0.2759449676001764e-9, (19, 5) = -0.3660513213351561e-11, (19, 6) = -0.8496643946552686e-10, (20, 1) = -0.6997436017168929e-13, (20, 2) = -0.4286845890806063e-11, (20, 3) = 0.49699435844552223e-10, (20, 4) = -0.15489670735342713e-9, (20, 5) = -0.8938460341045965e-12, (20, 6) = -0.8276448093060273e-10, (21, 1) = .0, (21, 2) = .0, (21, 3) = 0.44444299545291215e-10, (21, 4) = -0.11237818753716929e-9, (21, 5) = .0, (21, 6) = -0.829435465948473e-10}, 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 8 elif outpoint = "error" then return HFloat(9.8068495484932e-9) elif outpoint = "errorproc" then error "this is already the error procedure" elif outpoint = "rawdata" then return [6, 21, [f(x), diff(f(x), x), diff(diff(f(x), x), x), diff(diff(diff(f(x), x), x), x), g(x), diff(g(x), x)], 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(6, 'datatype' = 'float'[8]); evalhf(`dsolve/numeric/lagrange`(21, 6, X, Y, outpoint, var(yout), var(L), var(V))) else L := Vector(4, 'datatype' = 'sfloat'); yout := Vector(6, 'datatype' = 'sfloat'); `dsolve/numeric/lagrange`(21, 6, X, Y, outpoint, yout, L, V) end if; [x = outpoint, seq('[f(x), diff(f(x), x), diff(diff(f(x), x), x), diff(diff(diff(f(x), x), x), x), g(x), diff(g(x), x)]'[i] = yout[i], i = 1 .. 6)] 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 8 elif outpoint = "error" then return HFloat(9.8068495484932e-9) elif outpoint = "errorproc" then return eval(errproc) elif outpoint = "rawdata" then return [6, 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(6, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0}, datatype = float[8]); evalhf(`dsolve/numeric/hermite`(21, 6, 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(6, {(1) = 0., (2) = 0., (3) = 0., (4) = 0., (5) = 0., (6) = 0.}); `dsolve/numeric/hermite`(21, 6, X, Y, YP, outpoint, yout, L, V) end if; [outpoint, seq(yout[i], i = 1 .. 6)] end proc, (2) = Array(0..0, {}), (3) = [x, f(x), diff(f(x), x), diff(diff(f(x), x), x), diff(diff(diff(f(x), x), x), x), g(x), diff(g(x), x)], (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(x_bvp) elif outpoint = "sysvars" then return data[3] elif procname <> unknown then return ('procname')(x_bvp) else _ndsol := pointto(data[2][0]); return ('_ndsol')(x_bvp) end if end if; try res := solnproc(outpoint); [x = res[1], seq('[f(x), diff(f(x), x), diff(diff(f(x), x), x), diff(diff(diff(f(x), x), x), x), g(x), diff(g(x), x)]'[i] = res[i+1], i = 1 .. 6)] catch: error  end try end proc

(1)

plots:-odeplot(sol,[[x,f(x)],[x,g(x)]], x=-1..1, color=[red,blue]);

 

 


 

Download diffeq1.mw

If your goal is to prove the equality of 2 expressions (say A and B) then do  simplify(A - B)  or  is(A = B) :


 

restart;

Is_square := M[dmax]*(sigma^2*omega[rK]^2 + omega[r]^2)*L[sigma]/(3*p*omega[r]*omega[rK]*L[mu]^2*sigma^2);

(1/3)*M[dmax]*(sigma^2*omega[rK]^2+omega[r]^2)*L[sigma]/(p*omega[r]*omega[rK]*L[mu]^2*sigma^2)

(1)

Is_square2 := M[dmax]*(1 + omega[r]^2/(sigma^2*omega[rK]^2))*L[sigma]/(3*p*omega[r]*L[mu]^2/omega[rK]);

(1/3)*M[dmax]*(1+omega[r]^2/(sigma^2*omega[rK]^2))*L[sigma]*omega[rK]/(p*omega[r]*L[mu]^2)

(2)

simplify(%%-%);

0

(3)

 



Sometimes this requires additional conditions:

is(sqrt(x^2)=x) assuming x>=0;
                               
true

 

Download Qprime_20200621_new.mw

restart;
P:=proc(n)
local f;
f:=convert(series(exp(x^2),x=7.5,n+1), polynom);
evalf(Int(f, x=5..10));
end proc:

P(100);
evalf(Int(exp(x^2), x=5..10));  # Check

                            1.350882278*10^42
                            1.350882281*10^42


Addition. I took the expansion not in powers of  x , but in powers of  x-x0  where x0 = 7.5 (this is the middle of the range  5 ..10). If we take the expansion in powers of  x , then to obtain a satisfactory result, it is necessary to take a very large  

Download Int.mw


 

restart;
a:=rsolve({a(n+1)=sqrt(x*a(n)), a(1)=x}, a(n), makeproc);

proc (n) local L, i, val; options remember, system, `Copyright (c) 2003 by Waterloo Maple Inc.`; if not type(n, integer) then error "input must be an integer" elif n = 1 then x elif n < 1 then L := [x]; for i from 0 by -1 to n do val := traperror(L[1]^2/x); if val = lasterror then error "unable to compute recurrence for n<%1", i+1 end if; L := [val, op(1 .. -2, L)] end do; L[1] elif 1 < n then L := [x]; for i from 2 to n do val := traperror((x*L[1])^(1/2)); if val = lasterror then error "unable to compute recurrence for n>%1", i-1 end if; L := [op(2 .. -1, L), val] end do; L[-1] end if end proc

(1)

L:=[seq(a(n), n=1..10)];
simplify(L) assuming x>=0;
simplify(L) assuming x<0;
 

[x, (x^2)^(1/2), (x*(x^2)^(1/2))^(1/2), (x*(x*(x^2)^(1/2))^(1/2))^(1/2), (x*(x*(x*(x^2)^(1/2))^(1/2))^(1/2))^(1/2), (x*(x*(x*(x*(x^2)^(1/2))^(1/2))^(1/2))^(1/2))^(1/2), (x*(x*(x*(x*(x*(x^2)^(1/2))^(1/2))^(1/2))^(1/2))^(1/2))^(1/2), (x*(x*(x*(x*(x*(x*(x^2)^(1/2))^(1/2))^(1/2))^(1/2))^(1/2))^(1/2))^(1/2), (x*(x*(x*(x*(x*(x*(x*(x^2)^(1/2))^(1/2))^(1/2))^(1/2))^(1/2))^(1/2))^(1/2))^(1/2), (x*(x*(x*(x*(x*(x*(x*(x*(x^2)^(1/2))^(1/2))^(1/2))^(1/2))^(1/2))^(1/2))^(1/2))^(1/2))^(1/2)]

 

[x, x, x, x, x, x, x, x, x, x]

 

[x, -x, -I*x, (-1/2+(1/2)*I)*x*2^(1/2), -(1/2)*x*2^(1/4)*(-2+2*I)^(1/2), -(1/2)*x*2^(5/8)*(-(-2+2*I)^(1/2))^(1/2), -(1/2)*x*2^(13/16)*(-(-(-2+2*I)^(1/2))^(1/2))^(1/2), -(1/2)*x*2^(29/32)*(-(-(-(-2+2*I)^(1/2))^(1/2))^(1/2))^(1/2), -(1/2)*x*2^(61/64)*(-(-(-(-(-2+2*I)^(1/2))^(1/2))^(1/2))^(1/2))^(1/2), -(1/2)*x*2^(125/128)*(-(-(-(-(-(-2+2*I)^(1/2))^(1/2))^(1/2))^(1/2))^(1/2))^(1/2)]

(2)

 



It is obvious that for any x> 0   a(n) = x  for any n , so if x=2 then  a(100)=2 .
 

Download rsolve1.mw


 

restart;

sol:=W(x)=_C1*(cosh(alpha*x)-sinh(alpha*x))+_C2*(cosh(alpha*x)+sinh(alpha*x))+_C3*sin(alpha*x)+_C4*cos(alpha*x);
W:=eval(W(x),sol);

W(x) = _C1*(cosh(alpha*x)-sinh(alpha*x))+_C2*(cosh(alpha*x)+sinh(alpha*x))+_C3*sin(alpha*x)+_C4*cos(alpha*x)

 

_C1*(cosh(alpha*x)-sinh(alpha*x))+_C2*(cosh(alpha*x)+sinh(alpha*x))+_C3*sin(alpha*x)+_C4*cos(alpha*x)

(1)

F:=[sinh(alpha*x),cosh(alpha*x),sin(alpha*x),cos(alpha*x)]:
W1:=collect(W, F);
assign(([D1,D2,D3,D4]=~[coeffs(W1, F)])[]);

(-_C1+_C2)*sinh(alpha*x)+(_C1+_C2)*cosh(alpha*x)+_C3*sin(alpha*x)+_C4*cos(alpha*x)

(2)

D1, D2, D3, D4;

_C1+_C2, -_C1+_C2, _C3, _C4

(3)

 


 

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