1. Second Order Non-Linear

It's been a while since my course in ODEs and I was stumped by this question:

Find the general solution of:
$\displaystyle y''(x) + x^2y(x) = 0$
where
$\displaystyle 0<x<\infty$

I have no idea what to do and any CAS I plug this in returns Bessel Functions. Is there a more elementary solution?

2. Re: Second Order Non-Linear

Hey Haven.

Just out of curiosity what are the parameters of the Bessel function? The reason I ask is that depending on what the parameters are, you can simplify it a great deal - possibly into trig functions.

3. Re: Second Order Non-Linear

Okay, Maple returns the following

$\displaystyle y(x) =c_1\sqrt{x}\mathrm{BesselJ}(\frac{1}{4},\frac{1}{ 2}x^2)+C_2\sqrt{x}\mathrm{BesselY}(\frac{1}{4},(1/2)x^2)$

4. Re: Second Order Non-Linear

Well, back in the day, any solution involing Bessel functions would be called elementary

To get a solution by hand, try expanding the solution in a McLaurin series
$\displaystyle y(x)=y(0)+y'(0)x+\sum_{n\geq2}c_nx^n$, substitute in the diff. equation, and obtain a closed formula for the coefficients $\displaystyle (c_n)$.
This is displayed at length here.

5. Re: Second Order Non-Linear

The trig functions use order equal to 1/2. In conjunction with the advice above, you could also muck around and see if you can use the other properties and relate it to the trig functions by linking it to bessel functions with order equal to 1/2. I don't know if its worth the effort though for this particular problem.