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

1. Solve the initial value problem $$y' - 3 y = g(t), \quad \quad y(0) = 0,$$ 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 $$\phi' - 3 \phi = \delta(t), \quad \quad \phi(0) = 0.$$
   $\displaystyle \phi(t) =$
   
   
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 \ = \ $