Griffiths quantum harmonic oscillator derivation

[Syrus]

Homework Statement

I am unsure as to a step in Griffiths's derivation of the quantum harmonic oscillator. In particular, I am wondering how he arrived at the equations at the top of the second attached photo, from the last equation (at the bottom) of the first photo (which is the recursion relation approximated for large j).

Any help?

Homework Equations

The Attempt at a Solution

 

[voko]

Substitute the approx. solution into the approx. equation. Is it satisfied?

 

[Syrus]

Is there a way to derive the solutions besides via a fortiori means?

 

[voko]

The could be some clever way, but since the exact solution is known, the approx. solution could equally have been obtained from it.

EDIT: it actually seems very simple: since for every "next" term we divide the previous by j/2, one should expect (j/2)! in the denominator, and c/(j/2)! should be fairly obvious as a possible solution.