Analytic solution of the heat equation with a source term

Question

I have the heat equation with Dirichlet boundary conditions
$$u_t(t,x)=u_{xx}(t,x)+\sin(x)$$
$$u(t,0)=u(t,2\pi)=0$$
$$u(0,x)=u_0(x)$$
Now, without the source term I could write the solution as
$$u(t,x) = \sum_{n=1}^\infty B_n \sin(n\pi x)e^{-n^2\pi^2t}$$
where
$$B_n=2\int_0^{2\pi}u_0(x)\sin(n\pi x)dx$$
but I'm not sure what it looks like with a source term. I had a problem set question which assumed knowledge of the solution to show it converges to it's stationary form, so I'm guessing it can be derived using the homogenous equation.

Answer 1

Hint:

the $\sin x$ source term is time independent: what happens if you add $t \sin x$ to your solution?
does it respect the PDE ? and the boundary conditions ?