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If I have a system of differential equations, with coefficient matrix, A=$\begin{bmatrix} 1&13\\-2&-1\end{bmatrix}$. The eigenvalues are $\lambda= \pm 5i$. The phase portrait is a center with a clockwise direction field. How would I go about determining the equation of one of the rotated ellipse orbits?

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Equation for the phase paths: $$\frac {dx}{dy}=\frac {x+13y}{-2x-y}$$ $$({-2x-y}){dx}= ({x+13y})dy$$ $$-2x{dx}= ydx+xdy+13ydy$$ Note that $ydx+xdy=dxy$ : $$-2x{dx}= dxy+13ydy$$ Integrate: $$C=x^2+ xy+ \frac {13} 2 y^2$$ It's an ellipse.

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    $\begingroup$ Outstanding answer thanks! $\endgroup$ – user707991 Nov 23 '19 at 23:31
  • $\begingroup$ yw @user707991.... $\endgroup$ – LostInSpace Nov 23 '19 at 23:34

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