Substituting variables in xmaxima

[coomast]

Hello,

I have a problem using the program xmaxima. It involves the substitution of a new dependent and independent variable in an ordinary differential equation. Let me clearify this with an example of which we know the solution beforehand. So consider the following equation:

$$\frac{d^2y}{dx^2}+x \cdot y=0$$

Substituting $y=\sqrt{x}\cdot g(x)$ gives:

$$x^2 \cdot \frac{d^2g}{dx^2}+x \cdot \frac{dg}{dx}+\left(x^3-\frac{1}{4}\right) \cdot g=0$$

Substituting in this equation $x^3=t^2$, we get:

$$t^2 \cdot \frac{d^2g}{dt^2}+t \cdot \frac{dg}{dt}+\left(\left(\frac{2t}{3}\right)^2 -\left(\frac{1}{3}\right)^2\right) \cdot g=0$$

Which is a Bessel differential equation with solution:

$$g(t)=A\cdot J_{1/3}\left(\frac{2t}{3}\right)+ B\cdot Y_{1/3}\left(\frac{2t}{3}\right)$$

Transforming into the previous variables:

$$g(x)=A\cdot J_{1/3}\left(\frac{2}{3}x^{3/2}\right)+ B\cdot Y_{1/3}\left(\frac{2}{3}x^{3/2}\right)$$

and thus the solution to the original differential equation:

$$y(x)=\sqrt{x}\cdot \left[A\cdot J_{1/3}\left(\frac{2}{3}x^{3/2}\right)+ B\cdot Y_{1/3}\left(\frac{2}{3}x^{3/2}\right)\right]$$

Now the question is how does one do that in xmaxima?

best regards,

coomast

 

[Greg Bernhardt]

Hi Coomast,

I'm not sure how to use xmaxima to solve the given differential equation, but I can offer some advice. Have you tried researching online to see if anyone else has posted a solution to a similar problem using xmaxima? There may be other tutorials or helpful posts that could provide insight into how to solve this problem. Additionally, you could reach out to the developers of xmaxima directly and ask for help.

Good luck!

 

[oswaler]

Hello coomast,

Thank you for reaching out with your question about using xmaxima for substitution of variables in an ordinary differential equation. It seems like you have a good understanding of the process and have provided a clear example for clarification. In order to perform this substitution in xmaxima, you can use the 'subst' command. For your example, it would look like this:

subst([y=sqrt(x)*g(x),x=t^2], x^2*diff(g,x,2)+x*diff(g,x)+(x^3-1/4)*g=0);

This will give you the Bessel differential equation with the solution you provided. I hope this helps and please let me know if you have any further questions. Best regards,