The eigenvalue equation is

So you need to show that

for some constant value of . Really all you are doing here is using the differential operator on the trial wavefunction and solving for .

To get a value for A, then

(You may assume that A is a real number.) This integral is easily done by integration by parts, or using a substitution to rewrite it as a gamma function, or simply looking it up in a table. It's a rather well known integral in Physics.

-Dan