Suppouse that $f$ is of class $C^2$ (it's second derivative is a continous function)and $g$ is of class $C^1$. Also such that $g,g'\in PC(2\pi)$ and $f,f',f'''\in PC(2\pi)$ and satisfy the differential equation $$f''+\lambda f=g$$where $\lambda\neq n^2, n=1,2,3,...$.
Find the Fourier series of $f$ in terms of the Fourier coeficients of $g$ and prove that converges everywhere.
(if $f\in PC(2\pi)$ then $\int_{-\pi}^{\pi}f(x)dx=\int_c^{c+2\pi}f(x)dx \quad\forall c\in\mathbb{R}$)
I'm stuck with this prove. Any suggestions would be great!