Problem #7: Consider the below wave equation with the given conditions.
\begin{align*} 100 \frac{\partial^2 u}{\partial x^2} &= \frac{\partial^2 u}{\partial t^2}, \quad 0 < x < 4, \ t > 0, \\ u(0, t) &= u(4, t) = 0, \quad t > 0 \\ u(x, 0) = 0, \quad \left. \frac{\partial u}{\partial t} \right|_{t=0} &= 6x(4 - x) = \sum_{n=1}^{\infty} \frac{384}{\pi^3 n^3} \left[ 1 - (-1)^n \right] \sin(n\pi x/4), \quad 0 < x < 4. \end{align*} The solution to the above boundary-value problem is of the form
\begin{align*} u(x, t) = \sum_{n=1}^{\infty} g(n, t) \sin \left( \frac{n\pi}{4} x \right) \end{align*} Find the function $g(n, t)$.
