The method of eigenfunction expansions is often useful for nonhomogeneous
problems related to the wave equation or its generalizations. Consider the
problem
$$
r(x) u_{u}=\left[p(x) u_{x}\right]_{x}-q(x) u+F(x, t)
$$
$$
\begin{aligned} u_{x}(0, t)-h_{1} u(0, t)=0, & u_{x}(1, t)+h_{2} u(1, t)=0 \\\
u(x, 0)=f(x), & u_{t}(x, 0)=g(x) \end{aligned}
$$
This problem can arise in connection with generalizations of the telegraph
equation (Problem 16 in Section 11.1 ) or the longitudinal vibrations of an
elastic bar (Problem 25 in Section \(11.1) .\)
(a) Let \(u(x, t)=X(x) T(t)\) in the homogeneous equation corresponding to Eq.
(i) and show that \(X(x)\) satisfies Eqs. ( 28) and ( 29) of the text. Let
\(\lambda_{n}\) and \(\phi_{n}(x)\) denote the eigenvalues and normalized
eigenfunctions of this problem.
(b) Assume that \(u(x, t)=\sum_{n=1}^{\infty} b_{n}(t) \phi_{n}(x),\) and show
that \(b_{n}(t)\) must satisfy the initial value problem
$$
b_{n}^{\prime \prime}(t)+\lambda_{n} b_{n}(t)=\gamma_{n}(t), \quad
b_{n}(0)=\alpha_{n}, \quad b_{n}^{\prime}(0)=\beta_{n}
$$
where \(\alpha_{n}, \beta_{n},\) and \(\gamma_{n}(t)\) are the expansion
coefficients for \(f(x), g(x),\) and \(F(x, t) / r(x)\) in terms of the
eigenfunctions \(\phi_{1}(x), \ldots, \phi_{n}(x), \ldots\)
