## Quantum Harmonic Oscillator Differential Equation help

Hi, so i am looking at the quantization of the harmonic oscillator and i have the following equation...

ψ''+ (2ε-y$^{2}$)ψ=0

I am letting y$\rightarrow$ $\infty$ to get...

ψ''- y$^{2}$ψ=0

It says the solution to this equation in the same limit is...

ψ= Ay$^{m}$e$^{\pm y^{2}/2}$

The positive possibility in the exponential is ignored since it is not in the physical Hilbert space. My question is how did they solve this differential equation? I have read a couple websites and it says that you just have to "guess" it... however, is there a logical way to why you would guess this? Thank you
 PhysOrg.com science news on PhysOrg.com >> Ants and carnivorous plants conspire for mutualistic feeding>> Forecast for Titan: Wild weather could be ahead>> Researchers stitch defects into the world's thinnest semiconductor
 The solutions of this EDO are known in terms of Modified Bessel functions or alternately in terms of Parabolic Cylinder functions (in attachment) Attached Thumbnails
 Blog Entries: 9 Recognitions: Homework Help Science Advisor For the ODE just use the Frobenius method. Series expansion.

## Quantum Harmonic Oscillator Differential Equation help

Okay i understand, thank you very much