The equation presented is a partial differential equation (PDE) problem of the heat equation type with additional forcing terms. Here's how we can approach solving it:

The PDE is:

$u_t = 4u_{xx} + e^{-3t} \sin(2\pi x)$

with boundary conditions:

$u(0,t) = u(1,t) = 0, \quad u(x,0) = 0$

Steps to solve:

1. Solve the homogeneous part: First, solve the homogeneous PDE: $u_t = 4u_{xx}$ using separation of variables. We assume a solution of the form: $u(x,t) = X(x)T(t)$ Substituting this into the PDE, we get two separate ODEs for $X(x)$ and $T(t)$.

2. Apply boundary conditions: Use the boundary conditions $u(0,t) = u(1,t) = 0$ to find the eigenvalues and eigenfunctions for $X(x)$.

3. Solve the non-homogeneous part: The forcing term $e^{-3t} \sin(2\pi x)$ introduces a non-homogeneous component. We use the method of undetermined coefficients or a Green's function approach to solve for the particular solution corresponding to this term.

4. Combine solutions: The general solution will be the sum of the homogeneous solution and the particular solution.

Let me know if you'd like a more detailed breakdown of these steps or assistance with the solution process!