Math 6410 • ODE

Suppose \(D\) is the rectangle defined by
\(\lvert x-x_0\rvert\le a\), \(\lvert y-y_0\rvert\le b\) in the plane \(\R^2\). Suppose
that \(f\) is a real valued function defined on \(D\) which is continuous, hence bounded
by some nonnegative number \(M\) on \(D\).

\begin{enumerate}
\item[(i)] Show that if \(\alpha=\min\{a,b/M\}\), then there exists a function \(y(x)\),
defined on \(\lvert x-x_0\rvert\le\alpha\), such that
\[
y'(x)=f(x,y(x))\quad\text{and}\quad y(x_0)=y_0.
\]

\item[(ii)] Show that the solutions \(y\) guaranteed in part (i) may not be unique;
that is, give an example in which (i) holds and more than one solution \(y(x)\) exists
through the point \((x_0,y_0)\) in your example.
\end{enumerate}