Fourier series method for differential equations

Let $g$ be $2\pi$-periodic, continuous on $[-\pi ,\pi ]$ and have Fourier series

Let $f$ be $2\pi$-periodic and satisfy the differential equation

where $k \neq n^{2}$, $n = 1,2,3,\ldots$. Find the Fourier series of $f$ and prove that it converges everywhere.

Use the Fourier method to solve the heat conduction equation $\partial u/\partial t=\partial^2u/\partial x^2$ on the circle $\mathrm{T}$ with initial data $u(0,x)=v(x)$ (see Problem 607).