Compute the Laplace transform \(\mathcal{L}\{y\}\) of \(y = -2 \, \delta\left(t - 4\right) + 3 \, e^{\left(2 \, t\right)} + 2 \, \mathrm{u}\left(t - 5\right) \) by using a transform table.

Then show how the integral definition of the Laplace transform to obtains same result.

Answer:

\[ \mathcal{L}\{y\} = \frac{2 \, e^{\left(-5 \, s\right)}}{s} + \frac{3}{s - 2} - 2 \, e^{\left(-4 \, s\right)} \]