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I've just encountered the following differential equation in a design problem

$\dot{l}\cos \theta - \dot{\theta}l \sin \theta = A[\dfrac{1}{\sqrt{r}}-\dfrac{1}{\sqrt{r+l\sin \theta}}] \quad (*)$.

Here $A$ and $r$ are constant parameters. Regarding the meaning of the parameters, we know that $r \gg l$, thereby $r \approx r+l\sin \theta $. Thus, the RHS can be neglected which leaves us with the differential equation

$\dot{l}\cos \theta - \dot{\theta}l \sin \theta = 0$,

which is easy to be solved. ($\dfrac{\dot{l}}{l} = \dot{\theta}\tan \theta$, so $\ln{l} = \ln \dfrac{1}{|\cos \theta|}$, and finally $l = \dfrac{1}{|\cos \theta|}$).

If one relaxes the $r \approx r+l\sin \theta $ assumption, is there any closed solution to the equation $(*)$?

  • the l.h.s is $$\frac{d}{dt}l\cos\theta$$ which you can solve to give the later equation (assuming you fix the constant of integration. You could try to use a half angle approach to transform the trig components. – Chinny84 Jul 31 '19 at 12:53
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    @Chinny84: The problem is that I don't know how to compute a closed-form solution to the integral of the RHS. Thus, can you explain in detail how the integration of the LHS helps me? –  Jul 31 '19 at 13:04
  • You could try and transform the $\cos \theta$ and $\sin \theta$ using the half-tangent sub or something else. Then see if there is a closed form. – Chinny84 Jul 31 '19 at 13:45

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