Considering the general solution of the Laplace Equation we get Xn(x) = sin(nx) Yn(y) = Ancosh(ny) + Bnsinh(ny) Lets have Yn(0) = 1 and An = 1 for convienience. As we know Yn() = 0 , hence solving for Bn => Bn = -cosh(n)/sinh(n) => Yn(y) = sinh(n - ny)/sinh(n) Since f(x) = sin(4x) => u(x,y) = n=1 sin(4x) * ( sinh(n - ny)/sinh(n) ) Solution Considering the general solution of the Laplace Equation we get Xn(x) = sin(nx) Yn(y) = Ancosh(ny) + Bnsinh(ny) Lets have Yn(0) = 1 and An = 1 for convienience. As we know Yn() = 0 , hence solving for Bn => Bn = -cosh(n)/sinh(n) => Yn(y) = sinh(n - ny)/sinh(n) Since f(x) = sin(4x) => u(x,y) = n=1 sin(4x) * ( sinh(n - ny)/sinh(n) ).