I'm working through an example and I don't understand the last part:

We have the eigenvalue problem:

with

Show that zero is an eigenvalue if and only and find the corresponding eigenfunction.

Suppose that

Then

For y(x) = Ax to be a n eigenfunction we require A not equal to 0 which is only possible if .

Normalisation requires

So

I thought normalised required the function sqaured to be equal to 1. So I don't understand where the x^4 has come from and why it is not x^2.

Can anyone please explain?

Thanks in advance!