Math 6410 • ODE

Let \(E\) be an open subset of \(\R^n\). Let \(f\) be a continuously
differentiable function mapping \(E\) into \(\R^n\), and let \(x_0\) belong to \(E\).

\begin{enumerate}
\item[(a)] Use the method of successive approximations to prove that the initial
value problem \(x'=f(x)\), \(x(0)=x_0\) has a unique solution on some interval
\([-a,a]\), and give an estimate for \(a\).

\item[(b)] Suggest how the result in a) can be used to prove a similar result for the
nonautonomous case.
\end{enumerate}