Got this (rather tricky) separation of variables problem:

 

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

(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.

 

 

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!!!!

Ever so sorry, i've transcribed the first part of the question incorrectly. Part (a) should read

 

(a) Find the general solution to the equation d²X(x) / dx² = 0

 

...making me even more confused!