Substituting variables in xmaxima

1. Apr 11, 2009

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