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Letting $\vec{r}=\begin{pmatrix}x(\theta)\\y(\theta)\end{pmatrix}$ and that $\vec{v}^\perp$ denote a vector perpendicular to the vector $\vec{v}$. Then are there values of $c,d$, with $$\vec{r}''+c(\vec{r}'^\perp)+d{1\over |\vec{r}|^3}\vec{r}=0,$$ describing a circle for large enough $\theta$?

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What's the source of the problem? And you do realize that your definition of $\vec{v}^\perp$ doesn't fix $\vec{v}^\perp$ uniquely? –  Raskolnikov Apr 1 '12 at 19:21

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Assume that $v^\perp$ is $v$ turned counterclockwise. Then the vector $(\cos\theta, \sin\theta )$ satisfies the given equation provided that $-1-c+d=0$.

With the clockwise choice of orthogonal vector the condition becomes $-1+c+d$.

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