What's the solution of the differential equation $y^4 = k^2 (y^2 + y'^2\csc^2\alpha)$, where $y$ is a function of $x$ and $\alpha$ is a constant?
Polynomial solutions don't seem to work, because the LHS will always have higher degree than the RHS. Solutions of the form $A\cos(x\sin\alpha)+b\sin(x\sin\alpha)$ don't work either, but maybe something similar does?
This comes from the 1st integral of the Euler-Lagrange equation for the functional $\int{y^2 + y'^2\csc^2\alpha)^{1/2}}dx$, which is the arc-length of a curve $r(\theta)$ on a cone with interior angle $2\alpha$, where $y=r$ and $x=\theta$. Perhaps there's a more useful way of using the Euler-Lagrange equation, giving an ODE whose solution is obvious?