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\begin{equation} \sin(\theta + d\theta) = \sqrt{1 + \frac{dy}{y}}\cdot{\sin(\theta)} \end{equation} I think this is a non-linear and non homogeneous first order equation. I found this whilst trying to solve a problem on a maths puzzles website which I think is called the 'Baristochrone' problem. Having thought about it a while, I finally ended up getting this as the result, where $\theta$ is the angle of the mass as a function of the vertical distance of the mass from it's origin, $y$.

Does anyone know how to solve this equation? I'm really just trying to find the function $\theta$, so could the answer be simplified to avoid solving that equation?

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This equation is meaningless as it combines finite and differential quantities under nonlinear functions.

We fix that using the Taylor development and ignoring second order terms and higher, giving

$$\sin(\theta)+\cos(\theta)d\theta=(1+\frac{dy}{2y})\sin(\theta)$$

or

$$\cos(\theta)d\theta=\frac{\sin(\theta)dy}{2y}.$$

Now this is a separable equation that can be integrated as

$$\ln(\sin(\theta))=\frac12\ln(y)+C$$ or

$$y=C\sin^2(\theta).$$

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  • $\begingroup$ how did you get to $1+ dy/2y$ $\endgroup$ – Dis-integrating Apr 9 '16 at 13:29
  • $\begingroup$ @gebra: as said, by Taylor. $\endgroup$ – Yves Daoust Apr 9 '16 at 13:29
  • $\begingroup$ the taylor series expansion of sqrt(1 + dy/y)? $\endgroup$ – Dis-integrating Apr 9 '16 at 13:31
  • $\begingroup$ or was it binomial? $\endgroup$ – Dis-integrating Apr 9 '16 at 13:32
  • $\begingroup$ @gebra Taylor and binomial coincide on this development. $\endgroup$ – Yves Daoust Apr 9 '16 at 13:35

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