Math 6410 • ODE

\textbf{Problem.}
\begin{enumerate}
\item[(a)] Let \(D\) be an open set in \(\mathbb{R}\times\mathbb{R}^N\) and let
\[
  f:D\to\mathbb{R}^N
\]
be a continuous function such that \(f(t,x)\) satisfies a local Lipschitz condition with
respect to the \(x\) variable. Consider the initial value problem
\[
  x'=f(t,x),\quad x(t_0)=x_0,\quad (t_0,x_0)\in D.
\]
State what is known about the existence and continuability of solutions of this problem.
Be particularly detailed about the behavior of solutions as the time variable approaches
the endpoints of existence intervals.

\item[(b)] Apply the above results by sketching the solution picture for the scalar
differential equation
\[
  x'=\frac{1}{tx}.
\]
(I.e. find \(D\) and then consider typical cases.)
\end{enumerate}