I've been trying to solve the following second order PDE:
$\frac{d^2u}{dx^2} + \frac{d^2u}{dy^2} = u + 3$
With initial boundary conditions:
$u(0,y)=0$ and $u(1,y)=0$ for $0<y<1$
$u(x,0)=0$ and $u(x,1)=0$ for $0<x<1$
The first thing I did was trying to solve the homogeneous equation using separation of variables and I got:
$U_h=(C_1e^{\lambda x}+C_2e^{-\lambda x})(C_3e^{\sqrt{1+\lambda^2}x}+C_4e^{-\sqrt{1+\lambda^2}x})$.
But I am having troubles trying to solve the $U_p$ since I don't exactly know what to do with the right side of the equation, I'd appreciate any kind of help.
