How Can I Solve Second Order Inhomogeneous PDEs with a Forcing Term?

[Onias]

Homework Statement

Hello, I'm having a problem solving second order inhomogeneous PDEs, for example the standard heat equation with a forcing term Sin(x)Sin(t) added onto it on the right hand side.

Homework Equations

u_t = u_xx + Sin(ax)Sin(bt)

The Attempt at a Solution

I can solve it for the homogeneous case using separation of variables, but where do I go from there? Thanks in advance.

 

[hunt_mat]

Greens function?