By using separation of variables, solve Laplace's equation

$$\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0 \quad 0<x<1, \quad 0<y<1$$

subject to

$$\begin{array}{ll} u(0, y)=0 & 0 \leqslant y \leqslant 1 \\ u(1, y)=0 & 0 \leqslant y \leqslant 1 \\ u(x, 0)=0 & 0 \leqslant x \leqslant 1 \\ u(x, 1)=2 \sin (3 \pi x) & 0 \leqslant x \leqslant 1 \end{array}$$