Solve $y'' + 2y' + y = \begin{cases} 4e^t, & 0 \leq t < 1 \\ 0, &t \geq 1 \end{cases}$ where $y(0) = 0, y'(0) = 0$

Question

Hi can you help me solve this? I have no idea how to solve the initial problem by Laplace transform when it is a piecewise function.

Solve the initial value problem using Laplace transform on $$y'' + 2y' + y = \begin{cases} 4e^t, & 0 \leq t < 1 \\ 0, &t \geq 1 \end{cases}$$ where $y(0) = 0, y'(0) = 0$

Answer 1

\begin{align} (\mathcal Lf)(s) = {} & \int_0^\infty f(t) e^{-st} \, dt \\[8pt] = {} & \int_0^1 4e^t e^{-st} \, dt \\[8pt] & \text{and so on.} \end{align}