Let $g(t) = e^{3t}.$

1. Solve the initial value problem using the technique of integrating factors. (Do not use Laplace transforms.)

$\displaystyle y(t) =$

2. Use Laplace transforms to determine the transfer function $\phi(t)$ given the initial value problem
$\displaystyle \phi(t) =$

3. Evaluate the convolution integral $(\phi \ast g)(t) = \int_0^t \phi(t-w)\, g(w)\ dw$, and compare the resulting function with the solution obtained in part (a).

$\displaystyle (\phi \ast g)(t) = \int_0^t$ $dw \ = \$