Problem #7: Consider the below wave equation with the given conditions.
\begin{align*} 49 \frac{\partial^2 u}{\partial x^2} &= \frac{\partial^2 u}{\partial t^2}, \quad 0 < x < 6, \ t > 0, \\ u(0, t) &= u(6, t) = 0, \quad t > 0 \\ u(x, 0) &= 0, \quad \left. \frac{\partial u}{\partial t} \right|_{t=0} = 3x(6-x) = \sum_{n=1}^{\infty} \frac{432}{\pi^3 n^3} \{1 - (-1)^n\} \sin(n\pi x/6), \quad 0 < x < 6. \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 \frac{n\pi x}{6} \end{align*} Find the function $g(n, t)$.
