Maple Questions and Posts

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After I switched to using arrows = curve in the call to DEtools:-DEplot I found errors in some calls. 

Is there a workaround other than not using this option? As phase plot looks like better with this option. For now, I will remove this option.

Here is an example

restart;

ode:=diff(y(x),x)-1/(-x^2+1)^(1/2) = 0;
x_range:=-0.99 .. 0.99;
DEtools:-DEplot(ode,y(x),x =x_range,y = -1.6 .. 1.6,[y(0) = 0],arrows = 'curve')

Error, (in DEtools/DEplot/direction) cannot assign a complex value to a hardware float Array

But this works

restart;

ode:=diff(y(x),x)-1/(-x^2+1)^(1/2) = 0;
x_range:=-0.99 .. 0.99;
DEtools:-DEplot(ode,y(x),x =x_range,y = -1.6 .. 1.6,[y(0) = 0])

I will report this to Maplesoft as it looks like a bug to me. 

May be someone can find a workaround so I can use arrows = curve?

Maple 2023.2 on windows 10

``

restart;

292176

ode:=diff(y(x),x)-1/(-x^2+1)^(1/2) = 0;
x_range:=-0.99 .. 0.99;
DEtools:-DEplot(ode,y(x),x =x_range,y = -1.6 .. 1.6,[y(0) = 0],arrows = 'curve')

diff(y(x), x)-1/(-x^2+1)^(1/2) = 0

-.99 .. .99

Error, (in DEtools/DEplot/direction) cannot assign a complex value to a hardware float Array

restart;

292176

ode:=diff(y(x),x)-1/(-x^2+1)^(1/2) = 0;
x_range:=-0.99 .. 0.99;
DEtools:-DEplot(ode,y(x),x =x_range,y = -1.6 .. 1.6,[y(0) = 0])

diff(y(x), x)-1/(-x^2+1)^(1/2) = 0

-.99 .. .99

 

Download detools_deplot_arrows_curve_problem_NOV_10_2023.mw

I recently answered a question concerning the Lane-Emden equation (see here LaneEmden) where the main topic was about finding its numerical solution.

The generic form of the Lane-Emden equation with parameter n is

LaneEmden := n -> (Diff(xi^2*(Diff(theta(xi), xi)), xi)) = -theta(xi)^n * xi^2

      d   /  2 / d            \\             n   2
n -> ---- |xi  |---- theta(xi)|| = -theta(xi)  xi 
      dxi \    \ dxi          //                  


I have just realized that I missed a "small" point in the original question: the OP ( @shashi598 ) wrote
"[...] Maple never comes out of evaluating [the] analytical solution when n=5 [...] ".
The important point here is that this solution (at least for some initial conditions) is known and simple (in the sense it doen't involve any special function).

So I tried for a few hours to verify this claim, and ended wondering myself if it might not be right?

Could you please tell me (I guess @shashi598 would be interested too in your return) if the differential equation LaneEmden(5) can be solved formally?
TIA.

Emden_equation.mw


EDITED:
After a little research it seems that very specigic method are used to build the analytic solution of the LaneEmden(n) (n not equal to 0, 1 and 5): serie expansions, homotopy, Adomian decomposition for instance.
I wasn't capable to find how the solution for LaneEmden(5) have been got for the first time (iseems to be atthe end of the 19th century).

Hello everyone, 

Can someone help me how to use a Matrix in sum?

Thanks

Given a system of 2 first order ode's which are autonomous (i.e. the RHS does not explicitly depend on time), one can make phase plot in Maple using DEtools:-DEplot

I am trying to get my Maple output to look like I get with Mathematica. I am getting close, but there are two issues. First, here is what I get in Maple

restart;

ode1:=diff(x(t),t)=2*x(t)*y(t);
ode2:=diff(y(t),t)=1-x(t)^2-y(t)^2;

DEtools:-DEplot([ode1,ode2],[x(t),y(t)],
                           t=0..1,x = -4..4, y = -4..4,arrows=curve,linecolor=red,
                           numpoints =300,arrowsize='magnitude',scene=[x(t),y(t)]);