Math 6410 • ODE

Consider the first order differential equation
\[
  \frac{dx}{dt}=f(x,t),
  \tag{3}
\]
where \(f(x,t)\) is periodic in \(t\), \(f(x,t+1)=f(x,t)\) for all \(x\) and \(t\).
\begin{enumerate}
\item[(a)] Define the Poincare map for this differential equation.
\item[(b)] Suppose \(f(x,t)=x(A(t)-x)\), and \(0<\alpha<A(t)<\beta\). Prove that the
differential equation has at least one nontrivial periodic solution.
\end{enumerate}