# A Second Order Differential Equation

1. Oct 19, 2012

### cyunus

1. The problem statement, all variables and given/known data
Hi, this problem is from first chapter of Mathematical Methods of Physics by Mathews and Walker. (Problem 1-36, second edition)

Consider the differential equation $y'' - xy + y^3 = 0$ for large positive x.

a-) Find an oscillating solution with two arbitrary constants.

b-) Find a (non trivial) particular nonoscillating solution.

2. Relevant equations

$x \rightarrow \infty$ $Ai(x) \rightarrow \frac{e^{-\frac{2}{3}x^{\frac{3}{2}}}}{2 \sqrt{\pi}x^{\frac{1}{4}}}$ and $Bi(x) \rightarrow \frac{e^{\frac{2}{3}x^{\frac{3}{2}}}}{\sqrt{\pi}x^{\frac{1}{4}}}$
3. The attempt at a solution

If we assume y is small then we get airy equation and $Ai(x)$ is a solution while $Bi(x)$ is not a solution since for $x \rightarrow \infty { } Bi(x) \rightarrow \infty$. Also it should be noted that $Ai(x)$ does not have a root for large x therefore cannot be considered as an oscillated solution. Also WKB method didn't give me an oscillating solution but I can't say I know it well.

Thanks

Last edited: Oct 19, 2012