The edges of a square sheet of thermally conducting material are at x=0, x=L, y= -L/2 and y=L/2
The temperature of these edges are controlled to be:
T = T0 at x = 0 and x = L
T = T0 + T1sin(pi*x/L) at y = -L/2 and y = L/2
where T0 and T1 are constants. The temperature obeys Laplace's equation grad² T (x,y) = 0
(a) Find the general solution to the equation d²X(x) / dx² = 0
(b) Use the method of separation of variables to find the solution to Laplace's equation that objeys the boundary conditions. You'll need to consider a superposition of two solutions: one with a separation constant equal to 0 and a second for which the separation constant is nonzero.
The Attempt at a Solution
Expanding the LaPlace's equation, I get my two ODEs:
d²X / dx² - k²X = 0
d²Y / dy² + k²Y = 0
Because one of the boundary conditions has a sin(pi*x/L) term in, I've written the general solution for d²X / dx² - k²X = 0 as:
X(x) = Asin(kx) + Bcos(kx) (think that's right...)
However I'm having major problems with part (b)... my natural assumption would be that the entire general solution reads as:
T(x,y) = [Asin(kx) + Bcos(kx)][Ce^ky + De^ky]
However using that I can't zero enough terms given my boundary conditions to get anywhere. HELP!!!!