MAT244-2013S > Term Test 2

TT2 Question 2

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Jason Hamilton:

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.

Jeong Yeon Yook:
q2 part e) and bonus

Jeong Yeon Yook:
q2 part a) b) and c)

Jeong Yeon Yook:
#2 part d)

Jeong Yeon Yook:
part d) continued