eigenfunction expansion to represent the best solution

Use the appropriate eigenfunction expansion (if it exists) to represent the best solution of the following problems.

$$u”+ u =f(x)$$ with boundary condition $$u(0) = u(2pi), u'(0) = u'(2pi)$$

What I have now is trying to solve $$u”+u=lambda u$$
So the general equation is $$Acos(sqrt{1-lambda}x)+Bsin(sqrt{1-lambda}x)$$
Then using the boundary conditions to set a matrix, the eigenfunctions exists if only if $$cos(2pisqrt{1-lambda})=1$$Therefore $$2pisqrt{1-lambda} =2pi n $$So $$lambda = 1-n^2$$
and eigenfunction would be $$u_n = a_nsin nx+b_ncos nx$$

Am I on the right track ?
Is this simply the Fourier expansion ?