Kitonum

15082 Reputation

24 Badges

12 years, 26 days

MaplePrimes Activity


These are answers submitted by Kitonum


 

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)

 


 

Download collect_new.mw

The Iterator package was introduced in Maple 2016. If you have an older version of Maple, then you can use the commands  combinat:-nextperm  and  combinat:-nextcomb .

primpart(98-28*sqrt(7), sqrt(7));
content(98-28*sqrt(7), sqrt(7));
%*``(%%);
expand(%);

                                   

Or manually:

Expr:=98-28*sqrt(7);
d:=igcd(98,28);
d*``(Expr/d);

 

The integral of a periodic function is the same on any period-long interval. The contradiction below is due to limited accuracy in approximate calculation. With increasing accuracy, both methods yield consistent results. Infinity is explained by the fact that the function is discontinuous, its denominator is 0 at some points.


 

restart;
x(t) := -3.703703704*10^(-7)*(0.000111668023*cos(1000/33*sqrt(1122)*t) - 0.0001214712007*sin(1000/33*sqrt(1122)*t) - 0.0002325581396*sqrt(561)*sqrt(2)*(-0.0004467462845*sqrt(1122)*sin(1000/33*sqrt(1122)*t) + 0.0004467462845*sqrt(1122)*cos(1000/33*sqrt(1122)*t)))/((2.074226433*10^14*cos(1000/33*sqrt(1122)*t) + 2.074226433*10^14*sin(1000/33*sqrt(1122)*t))*(4.895037587*10^(-11) + 0.01685634229*(0.00001474262739*cos(1000/33*sqrt(1122)*t) + 0.00001474262739*sin(1000/33*sqrt(1122)*t))^2)^2);
P:= 33*Pi/1000/sqrt(1122):

# Digits=10
RMS:= evalf(sqrt(1/2/P*Int(x(t)^2, t= -P..P)));
RMS:= evalf(sqrt(1/2/P*Int(x(t)^2, t= 0..2*P)));

Digits:=20:
RMS:= evalf(sqrt(1/2/P*Int(x(t)^2, t= -P..P)));
RMS:= evalf(sqrt(1/2/P*Int(x(t)^2, t= 0..2*P)));
 

-0.3703703704e-6*(0.111668023e-3*cos((1000/33)*1122^(1/2)*t)-0.1214712007e-3*sin((1000/33)*1122^(1/2)*t)-0.2325581396e-3*561^(1/2)*2^(1/2)*(-0.4467462845e-3*1122^(1/2)*sin((1000/33)*1122^(1/2)*t)+0.4467462845e-3*1122^(1/2)*cos((1000/33)*1122^(1/2)*t)))/((0.2074226433e15*cos((1000/33)*1122^(1/2)*t)+0.2074226433e15*sin((1000/33)*1122^(1/2)*t))*(0.4895037587e-10+0.1685634229e-1*(0.1474262739e-4*cos((1000/33)*1122^(1/2)*t)+0.1474262739e-4*sin((1000/33)*1122^(1/2)*t))^2)^2)

 

0.3200233260e-5

 

Float(infinity)

 

Float(infinity)

 

Float(infinity)

(1)

 


 

Download Infinity.mw

To plot surfaces of revolution it is convenient to use their parametric equations in which to use polar coordinates on the plane. The 1 ex. :

restart;
x:=r*cos(phi): y:=r*sin(phi):
plot3d([[x,y,r],[x,y,4]], r=0..4, phi=0..2*Pi, style=surface, color=grey, scaling=constrained, labels=["x","y",z]);

                        

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