Math 6410 • ODE

Consider the differential equation
\[
\frac{dx}{dt}=f(x,t),\quad x\in\mathbb{R}^n,
\]
with initial condition \(x(0)=x_0\).
\begin{enumerate}

\item
State a theorem, with conditions on \(f(x,t)\), guaranteeing the existence and uniqueness
of solutions of the differential equation on some interval \(0<t<T_0\).


\item
Outline the proof of your theorem, indicating the main steps of the proof (without all
the details), indicating where your stated assumptions on \(f(x,t)\) are invoked. What
are Picard iterates and how do they play a role in your proof?

\end{enumerate}