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!