Math 6410 • ODE

Let \(E\) be an open subset of \(\R^n\) containing \(x_0\) and assume that \(\mathbf{f}\)
is a continuously differentiable function mapping \(E\) into \(\R^n\).
\begin{enumerate}
\item Use the method of successive approximations to prove that the initial value
problem \(x'=\mathbf{f}(x)\), \(x(0)=x_0\), has a unique solution on some interval
\([-a,a]\), and give an estimate for \(a\).
\item Indicate how the result in (a) can be used to prove a similar result for the
nonautonomous case.
\end{enumerate}