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