Exercise 2: (25 pts) Solve the following wave equation on [0, ?]
$\begin{cases}
u_t(x, t) = u_{xx}(x, t), \\
u_x(0, t) = 0, \\
u_x(\pi, t) = 0, \\
u(x, 0) = f(x), \\
u_t(x, 0) = 0.
\end{cases}$
Exercise 3: (25pts) Solve the following heat equation on [0, 1]
$\begin{cases}
u_t(x, t) = u_{xx}(x, t), \\
u(0, t) = 0, \\
u(1, t) = t, \\
u(x, 0) = 0.
\end{cases}$
Note: This heat equation has a non-homogeneous boundary condition.
