Analytic derivation of quantum SHO

[Aziza]

In Griffith's derivation of the quantum SHO, he uses some funny math:

first he considers asymptotic behavior to get ψ=Ae-(ε^2/2)
then he 'peels off the exponential part' to say that ψ=h(ε)e-(ε^2/2)
then he hopes that h(ε) will have simpler form than ψ(ε)

I can kind of understand the first part, but I have no clue what he means by the second part, i don't understand the motivation for this step...given this ODE, i would not know to proceed this way.
and idk what reason we have to 'hope' that h(ε) will be simple, just from the above data.

I am not familiar at all with this method of solving differential equation and i cannot find any resource on it...does anyone know of a better explanation? everything i have found merely copies word for word griffith's derivation.

I also have liboff's and mahan's book and they are even worse at this explanation

 

[voko]

This argument is very similar in nature to the Euler-Lagrange method of variation of parameters. Instead of solving the original equation, we first solve some simplified equation. Then we assume that the solution to the original equation should be the same, except some of what used to be constants (parameters) in the approximate solution are now functions. So we plug our guesswork into the original equation and obtain equations for the new functions, which we hope will be simpler than the original equation.

 

[vela]

Mathews & Walker mentions this technique briefly, but I doubt you'll find their treatment any more satisfying.

Liboff doesn't even talk about solving the differential equation this way, so I'm not sure what you are referring to. He does look at the asymptotic behavior in his discussion of what the solutions should look like qualitatively, but to find the actual solutions, he uses an algebraic approach with the creation and annihilation operators.