I'm trying to solve this problem but having some difficulty, could somebody please help me out?

y" + (1+ey)y = 0. Subject to y(0)=1, y'(0)=0

I've used the expansion y(x)=y0(x) + ey1(x) + ...

when i differentiate this twice and substitute into my eqn and collect powers of e i get two equations:

y0" + y0 = 0

y1" + y1 = 0

I think this is wrong and not really sure what to do.

Thanks for any help (and sorry about the poor layout of the problem)