5. Vibration of a string of length $\pi$ that is subject to tension and damping is governed by
$\frac{\partial^2 u}{\partial t^2} + 2\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}$, $0 < x < \pi$, $t > 0$,
(5)
where $u(x, t)$ is the transverse displacement of the string at a position $x$ along the string
and time $t$. The ends of the string are fixed, hence the boundary conditions are
$u(0, t) = u(\pi, t) = 0$, $t > 0$.
(6)
Suppose that the initial displacement and velocity are
$u(x, 0) = 2\sin x$, $u_t(x, 0) = 0$, $0 < x < \pi$,
(7)
respectively.
Use separation of variables to solve equation (5) subject to boundary conditions (6) and
initial conditions (7).
If applicable, you may use the result that the eigenvalues of the boundary-value problem
$X'' - \lambda X = 0$, $X(0) = 0$, $X(\pi) = 0$,
are $\lambda_n = -n^2$ for $n = 1, 2, \dots$, with corresponding eigenfunctions
$X_n(x) = \sin nx$.
[TOTAL 14 marks]
