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**Final Exam / FE-P6**

« **on:**December 14, 2018, 08:06:54 AM »

**Typed solutions only. Upload only one picture (a general phase portrait; for general one can use computer generated)**

For the system of ODEs

\begin{equation*}

\left\{\begin{aligned}

&x' = 2y(x^2+y^2+4)\, , \\

&y' = -2x (x^2+y^2-16)

\end{aligned}\right.

\end{equation*}

**(a)**Find stationary points.

**(b)**Linearize the system at stationary points and sketch the phase portrait of this linear system.

**(c)**Find the equation of the form $H(x,y) = C$, satisfied by the trajectories of the nonlinear system.

**(d)**Sketch the full phase portrait.

*Hint*: avoid redundancy: