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**Term Test 2 / TT2 Question 2**

« **on:**March 27, 2013, 10:02:33 PM »

Consider the second order equation

\begin{equation*}

x''=x^4-5x^2+4

\end{equation*}

(a) Reduce to the first order system in variables $(x, y, t)$ with $y = x'$, i.e.

\begin{equation*}

\left\{ \begin{array}{ll}

x'=\ldots\\

y'=\ldots\\

\end{array}\right.

\end{equation*}

(b) Find solution in the form $H(x,y)=C$.

(c) Find critical points and linearize system in these points.

(d) Classify the linearizations at the critical points (i.e. specify whether they are nodes, saddles, etc., indicate stability and, if applicable, orientation) and sketch their phase portraits.

(e) Sketch the phase portraits of the nonlinear system near each of the critical points.

(f) Sketch the solutions on $(x,y)$ plane.