Can the ODE \psi''-y^2\psi=0 be solved using a general method?

[Luke Tan]

TL;DR

How do i solve this ODE?

When reading through Shankar's Principles of Quantum Mechanics, I came across this ODE

$$\psi''-y^2\psi=0$$
solved in the limit where y tends to infinity.

I have tried separating variables and attempted to use an integrating factor to solve this in the general case before taking the limit, but they didn't work.

I also tried to guess a solution of the form $e^{f(y)}$, and it quickly became clear that $$f(y)=\frac{y^2}{2}$$, but it feels like my guess is unmotivated and i didn't get the $y^m$ term since i didn't guess it would be there.

Is there any general method for this kind of ODE?

 

[phyzguy]

Try this:

https://en.wikipedia.org/wiki/Parabolic_cylinder_function

 

[Engie]

You could try to expand in Laurent series at infinity and factor out $y^m$...