Solving Laplace Equations using this boundary conditions?

[astrodeva]

The equation is Uxx + Uyy = 0
And domain of solution is 0 < x < a, 0 < y < b
Boundary conditions:
Ux(0,y) = Ux(a,y) = 0
U(x,0) = 1
U(x,b) = 2

What I've done is that I did separation of variables:
U(x,y)=X(x)Y(y)

Plugging into the equation gives:
X''Y + XY'' = 0

Rearranging:
X''/X = -Y''/Y = k

For case k > 0, I saw that it gives no non-trivial solutions.
For case k = 0, I solved it and found U(x,y) = y/b + 1

For case k < 0, I'm slightly lost.
X'' + kX = 0
Y'' - kX = 0

Using the X boundary conditions:

Using the Y boundary condition:

Using Fourier Series to find the coefficient:

But the integral just gives Dn = 0, and this doesn't satisfy u(x,0) = 1.

Can someone explain where I went wrong?

Thanks!

 

[pasmith]

Your working is confusing.

If you start with $X'' = kX$ and want $k < 0$, you should then be writing $\sin(\sqrt{|k|}x)$ and so forth, or defining $k = -c^2$ where $c \geq 0$.

You want to be using cosines, which in fact you end up doing, but only after expressly stating that $X_n = A_n\sin(n\pi x/a)$.

Now the cosine expansion of $1$ on $[0,a]$ is $\cos(0x)$. Thus you have correctly determined that all the other coefficients vanish.