Applying Initial Conditions

[MathematicalPhysics]

I need to find all the separated solns of

x^2 \frac{\partial^2 u}{\partial x^2} + x\frac{\partial u}{\partial x} + \frac{\partial^2 u}{\partial y^2} = 0

in the strip {(x,y) : 0 < y < a, -\infty < x < \infty }
the separated solns must also satisfy u = 0 on both the edges, that is, on y=0 and y=a for all values of x.

Iv got the general solutions to be..

X(x) = Dlnx + C , (k = 0)
X(x) = Dx^{n} + Cx^{-n} , (k \neq 0)

and

Y(y) = A\cos{ky} + B\sin{ky} , (k \neq 0)
Y(y) = Ay + B , (k = 0)

where k is just the constant iv let the two bits equal when I separated the variables. (well -k^2 actually).

I just need help interpreting the conditions to sort out the constants..I think!

[Galileo]

I haven't checked your answer, but if it is correct then, since u(x,y)=X(x)Y(y), the boundary conditions say:

u(x,0)=X(x)Y(0)=0
and
u(x,a)=X(x)Y(a)=0

So Y(0)=Y(a)=0

For example: if k=0, then applying the boundary condition at y=0 gives:
Y(0)=B=0