This problem involves solving the Initial Boundary Value Problem (IBVP) for the given Partial Differential Equation (PDE) using separation of variables and undetermined coefficients. Here's a breakdown of the solution approach:

Problem Setup

PDE:

$u_{tt} = c^2 u_{xx} + e^{-t} \sin\left(\frac{3\pi x}{2L}\right) - 4 \sin\left(\frac{7\pi x}{2L}\right),$ for $0 < x < L, 0 < t < \infty$.

Boundary Conditions (BCs):

$u(0, t) = 0, \quad u(L, t) = 0, \quad \text{for } 0 < t < \infty.$

Initial Conditions (ICs):

$u(x, 0) = 3 \sin\left(\frac{5\pi x}{2L}\right), \quad u_t(x, 0) = 0, \quad \text{for } 0 \leq x \leq L.$

Steps to Solve

1. Separation of Variables:

Expand $u(x, t)$ and the forcing function $Q(x, t)$ in terms of eigenfunctions of the spatial operator $u_{xx}$ under the boundary conditions.

The eigenfunctions for $u_{xx}$ with boundary conditions $u(0, t) = u(L, t) = 0$ are: $\phi_n(x) = \sin\left(\frac{n\pi x}{L}\right), \quad n = 1, 2, 3, \dots.$

The eigenvalues are: $\lambda_n = \left(\frac{n\pi}{L}\right)^2.$

2. Forcing Function Expansion:

Expand $Q(x, t) = e^{-t} \sin\left(\frac{3\pi x}{2L}\right) - 4 \sin\left(\frac{7\pi x}{2L}\right)$ in terms of $\phi_n(x)$: $Q(x, t) = e^{-t} \phi_3(x) - 4\phi_7(x),$ since $\frac{3\pi}{2L}$ and $\frac{7\pi}{2L}$ correspond to eigenfunctions with $n = 3$ and $n = 7$, respectively.

3. Solution Ansatz:

Write $u(x, t)$ as: $u(x, t) = \sum_{n=1}^\infty X_n(t) \phi_n(x),$ where $X_n(t)$ are the time-dependent coefficients.

4. Substitute into PDE:

Substituting $u(x, t)$ into the PDE yields: $\sum_{n=1}^\infty \left[X_n''(t) + c^2 \lambda_n X_n(t)\right] \phi_n(x) = Q(x, t).$

Match coefficients of $\phi_n(x)$: $X_n''(t) + c^2 \lambda_n X_n(t) = Q_n(t),$ where $Q_n(t)$ are the Fourier coefficients of $Q(x, t)$: