Consider the following initial value problem, defined for $t \geq 0$: $$\displaystyle\frac{dy}{dt} = t \ast t^{2}, \quad \quad y(0) = 2.$$

1. Find the Laplace transform of the solution.
   
   $\displaystyle Y(s) = {\mathcal L}\left\lbrace y(t) \right\rbrace =$
   
   
2. Obtain the solution $y(t)$.
   
   $y(t) =$