2. The voltage $V(x, y)$ at any point on a rectangular metal plate whose sides are given by $x = 0$, $x = 5$ and $y = 0$, $y = 15$, satisfies Laplace's equation
$\frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2} = 0$.
(7)
Suppose that the plate is earthed along the sides $x = 0$, $x = 5$ and $y = 0$ and that a voltage $f(x)$ is applied to the side $y = 15$ giving the boundary conditions
$V(x, 15) = f(x)$, $V(x, 0) = 0$, $V(0, y) = 0$, $V(5, y) = 0$.
(8)
(a) Use separation of variables to find a series solution for $V(x, y)$ at any point $(x, y)$ in terms of undetermined coefficients.
(b) If the voltage $f(x)$ applied along the side $y = 15$ is
$f(x) = \begin{cases} x, & 0 \le x \le \frac{5}{2} \\ 5 - x, & \frac{5}{2} \le x \le 5 \end{cases}$
(9)
determine the coefficients in the series solution.
