Let U(x,y)=X(x)Y(y), which satisfies the partial equation,

d^2u/dx^2 + d^2u/dy^2 = 0 [This is the Laplace equation]

Given U(0,y)=U(a,y) = U(x,0) = 0 and U(x,b)=g(x)

Show that U(x,y) = Sigma (from n=1 to infinity) Cn Sin(n*pi*x/a)sinh(n*pi*y/a)

Where n,a and b are constants.

My trouble is not how I start it but where does my Sinh(n*pi*y/a) come from??

I got two equations they are X''(x)-G*X(x)=0 and Y''(y)+G*Y(y)=0

Now Ignore solving the first equation because I already got CnSin(...)

My problem is Y''(y)+G*Y(y)=0 and getting some how sinh(n*pi*y/a)

Keep in mind that G=(n*pi/a) which is my eigenvalue.

Thanks