Math 6410 • ODE

\begin{enumerate}

\item
What is the statement of Gronwall's inequality? Where to start: Suppose \(\psi (x)\) satisfies
the inequality
\[
\psi(t)\le \alpha+\int_0^t \beta\psi(s)\,ds,
\]
then \(\cdots\).


\item
Suppose \(f(t,x)\in C(U,\mathbb{R}^n)\) is Lipschitz continuous in its second
argument, and let \(x(t)\) and \(y(t)\) be solutions of the same differential
equation \(\frac{dx}{dt}=f(t,x)\) with different initial data
\(x(0)=x_0\), \(y(0)=y_0\). Prove that

\[
|x(t)-y(t)|\le |x_0-y_0|\exp(Lt),
\]


for some positive constant \(L\) and some positive time \(0 \le t < T\). What is an estimate
for the constant \(L\)?


\item
Why may the inequality fail to hold for all time? Give an example of where this fails.

\end{enumerate}