Existence, Uniqueness & Bifurcations · Medium problem

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