Laplace's equation on a rectangle (mixed bndy)

[lordofspace]

Homework Statement

I'm having issues with a deceptively simple Laplace problem. If anybody could point me in the right direction it would be fantastic.

It's just Laplace's equation on the square [0.1]x[0,1] (or any rectangle you like) with a mixed boundary.

Homework Equations

U_xx+U_yy=0
U_x(0,y)= a (some constant)
U_x(1,y)= 0
U(x,0)=0
U_y(x,1)=0

The Attempt at a Solution

The main thing I've tried is just an ordinary seperable solution, but it's definitely not seperable. I need something different.

 

[foxjwill]

Actually, it is separable. Just let U(x,y)=A(x)B(y).

EDIT: It seems to me that if a isn't zero, B(y) must be a constant function since by the first condition,
$$U_x(0,y)=A'(0)B(y)=a \Rightarrow B(y)=\frac{a}{A'(0)}=\text{constant}.\qquad (1)$$
Further, by the third condition, either A(x)=0 for all x or B(0)=0. If A(x)=0, then clearly U(x,y)=0. If B(0)=0 (and a is nonzero), then by (1), B(y)=0 for all y, so U(x,y)=0. Thus, in order for U(x,y) to be non-trivial, a must be zero.