Existence, Uniqueness & Bifurcations · Medium problem

Apply Picard iteration to $y' = ty$, $y(0) = 2$: $\varphi_0(t) = 2$ and $\varphi_{n+1}(t) = 2 + \displaystyle\int_0^t s\,\varphi_n(s)\,ds$. Evaluate $\varphi_2(2)$.