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$$\frac{\mathrm d^2y}{\mathrm dt^2} + p\frac{\mathrm dy}{\mathrm dt} + qy = 0$$

If the eigenvalues are complex, what conditions on $p$ and $q$ guarantee that solutions spiral around the origin in a clockwise direction ?

I found the eigenvalues to be: $\lambda = -\frac{p}{2} \pm \frac{\sqrt{p^2 - 4q}}{2}$

and not sure how we can answer this. All I note that if $p > 0$ and $p^2 - 4q < 0$, then we have a spiral sink. How does it help me if I can get conditions for spiral sink or source and match their directions (counterclockwise or clockwise) when the problem indicates we are able to do this without the use of technology sketching a phase portrait?

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    So does this mean that we need p^2 - 4q < 0 ? This is when we get a spiral sink, and sinks are such that spiral around the origin in a clockwise direction, right?2011-07-25
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    @mary: Please edit in your question that you mean when $y$ results in a spiral around the origin *in its phase portrait*, as opposed to in the complex plane. Whether or not $y$'s image in $\mathbb{C}$ spirals clockwise or counterclockwise (or at all) into $0$ depends not just on the ODE's coefficients but on the initial conditions.2011-07-25
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    Ah - I eventually figured you were talking about the phase space, due to Robert's answer, but apparently my first impression was correct that you're talking about the complex plane.2011-07-25

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