Math 6410 • ODE

Consider the differential equation \(\ddot{x}+\eta\dot{x}+x^3-x=0\), with
\(0<\eta<2\sqrt{2}\) a constant.
\begin{enumerate}

\item
Sketch a phase portrait for this equation. Identify all fixed points and their type.


\item
Find a first integral of the motion if \(\eta = 0\). Make a sketch of these integral curves in
the \(x-\dot{x}\) plane.



\item
Suppose that at time \(t=0\),
\(\frac{\dot{x}^2}{2}+\frac{1}{4}x^4-\frac{1}{2}x^2<0\), and
\(0<x<\sqrt{2}\). Determine \(\lim_{t\to\infty}x(t)\), and explain how to
prove this.

\end{enumerate}