Exercise 4

Consider the initial value problem $$\begin{aligned} y' & = 2ty + t \\ y(0) & = 1. \end{aligned}$$

1. Use the fact that $y' = 2ty + t$ is a first-order linear differential equation to find a solution to the initial value problem.

2. Let $\phi_0(t) = 1$ and use Picard iteration to find $\phi_n(t)$.

3. Show that the sequence $\{ \phi_n(t) \}$ converges to the exact solution that you found in part (a) as $n \to \infty$.