Chapter 12.6, Problem 9E

In Problems 1-12 proceed as in Example 1 to solve the given boundary-value problem.

9. When a vibrating string is subjected to an external vertical force that varies with the horizontal distance from the left end, the wave equation takes on the form

$a^2\frac{{\partial }^{2}u}{\partial x^2}+Ax=\frac{{\partial }^{2}u}{\partial t^2}$,

where A is a constant. Solve this partial differential equation subject to

u(0, t) = u₀, u(1, t) = 0, t > 0

u(x, 0) = 0, ${\frac{\partial u}{\partial t}|}_{t=0}=0$, 0 < x < 1.