Are you trying to find the Green's function for this problem?

The Green's function, G(x,y), for this problem must satify these conditions:

1) for all .

2) G(0,y)= G(1,y)= 0 for all y between 0 and 1.

3) where is the limit of G'(x,y) with respect to x as x approaches y from above and is the limit as x approaches y from below.

Find the solutions to (1) that satisfy G(0,y)= 0 and solutions that satisfy G(1,y)= 0. You will have two functions each with an unknown constant. Use the fact that they must match at x= y and (3) to determine those two constants.