Math 6410 • ODE

\begin{enumerate}
\item[(a)] Let \(f(t,x):\R\times\R^n\to\R^n\) be a continuously differentiable
function. Show that there is an \(\epsilon>0\) and a unique continuously differentiable
function \(y(t)\in\R^n\) for \(t\in[0,\epsilon]\) that satisfies the initial value
problem
\[
  \frac{dx}{dt}=f(t,x),\qquad x(0)=x_0.
\]
\item[(b)] Construct the first few Picard iterates for the differential equation
\[
  \dot{x}=2t-\sqrt{\max(0,x)},\qquad x(0)=0.
  \tag{1}
\]
Show that the Picard iterates do not converge. Why do they not converge?
\end{enumerate}