Solve BVP using Eigenfunction Expansion

[bitty]

Homework Statement

Solve BVP y''[x]-y[x]=Cos[2x] using eigenfunction expansion.
We know y'[0]=y'[Pi]=0

Homework Equations

Fourier it up

The Attempt at a Solution

Is the corresponding Sturm-Liouville problem:
f''[x]+/lambda*f=0?

All the examples we've done have been of form y''[x]+y[x]=g[x] but does a negative sign in front of y[x] change the corresponding S/L problem we use?

It follows that f[x]=Cosnx

Because I proceed to solve for y[x]=Sum[A_n*f[x]] and I find the coefficents and all but when I plot my Fourier expansion for y[x], it is very different from the actual result of y[x]. I'm guessing I wasn't supposed to use f''[x]+/lambda*f=0 to find my eigenfunction expansion, but I'm confused as to why I'm getting the wrong answer

 

[dikmikkel]

Edit: sorry i misunderstood