I need to solve the following BVP:

$\Delta u - 1/\delta \* u = R(x,y)$, 

where $\Delta$ is the laplacian operator. and boundary conditions:

$u(0,y)=u(L,y)=u(x,0)=u(x,L)=0$

where $L=1$, $\delta=0.01$, and $R(x,y)=1$. 

I understand the basic concept of separation of variables, i.e. rewriting $u(x,y)$ as the product of some function of $x, X(x)$, and some function of $y, Y(y)$. Then by plugging $u(x,y)=X(x)Y(y)$ into the PDE, obtaining two ODEs that are much easier to solve. My solution started as...

$X''Y+XY''-1/\delta\*XY=0$

$X''/X=(1/\delta\*Y-Y'')/Y=-\lambda^2$

$X(x) = Asin\lambda x + Bcos\lambda y$

$Y(y) = C\_1 e^{\sqrt{1/\delta+\lambda^2}y} + C\_2 e^{-\sqrt{1/\delta+\lambda^2}y}$

eventually...

$u(x,y) = \sum{A\_n(e^{\sqrt{1/\delta+\lambda^2}y} + e^{-\sqrt{1/\delta+\lambda^2}y})sin{(n \pi x)}}$

However, this was merely for the homogeneous case, not the BVP as given. And in addition, it seemed counter intuitive to me that the functions $X(x)$ and $Y(y)$ should be of different forms. Why would $X(x)$ be composed of sinusoidal functions and $Y(y)$ be compased of exponential terms if $u(x,y)$ is taken to the second partial derivative in both the x and y directions?