# I How do I solve this ODE?

#### Luke Tan

Summary
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?

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Try this:

#### Engie

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