I have the following PDE

$\displaystyle PDE: u_t=A^2(u_xx+u_yy)+f(x,y,t) $ such that A is a constant

$\displaystyle BC: u_x(1,y,t)=0$

$\displaystyle u_x(0,y,t)=0$

$\displaystyle u_y(x,1,t)=0$

$\displaystyle u(x,0,t)=0$

$\displaystyle IC: u(x,y,0)=0$

I've set up my ODEs using seperation of variables to get

$\displaystyle X''/X=k_1

Y''/Y=k_2

T'/(A^2)T=k_1+k_2$

where k_1 and k_2 are constants.

How do I account for my source term (f(x,y,t))? I'm reading up on eigenfunction expansion but so far it's only for the dimensional case.