Suppose I want to find a function $(x,z) \mapsto y((x^2+z^2)^{1/2})$ which must represent the interface between two translucent media with different refractive index and which must yield a perfect lens, i.e. a lens without spherical abberation: light rays coming in from $y=-\infty$ and propagating parallel with the $y$-axis must be refracted at the interface in accordance with Snell's law and the outgoing light ray must hit the $y$-axis in a certain point $F=(0,c,0) \in \mathbb{R}^3$ with the "focus" $F$ independent of whichever incoming light ray we considered (this latter property is what makes the lens "perfect"). Suppose we agree to choose WLOG that $y(0)=0$. After some elementary geometry, I find that the problem reduces to the ODE $$\frac{x}{(x^2+[c-y(x)]^2)^{1/2}} = \sin(\alpha(x))\cos(\theta(x)) - \sin(\theta(x))\cos(\alpha(x))\\ \sin(\alpha(x)) = \frac{y'(x)}{(1+[y'(x)]^2)^{1/2}}=n\sin(\theta(x)) \qquad(1)$$ with $n>1$ the refractive index of the medium above the interface relative to the medium below. I believe giants such as Leibniz, Newton, Huygens must have arrived at this point and gotten stuck in finding a closed-form solution to this ODE. Browsing the web I found a paper claiming a closed-form solution (at least to a similar problem) and reading the contents of their article they claim the solution involves a simple expression without reverting to the use of non-elementary functions. I'm skeptical though, so therefore the following question:
Q: Can solutions of the ODE (1), subject to the initial condition $y(0)=0$, be expressed in terms of elementary functions? What is the most transparent account of the solutions to this ODE?
(the 2nd question can be understood in the same vein as how the Kepler problem may be difficult to solve with e.g. time as the independent variable, but if one promotes a related variable -namely the angle- as the independent variable a closed-form solution in terms of elementary functions is easily obtained.)
(Apparently the problem of designing abberation-free lenses is colloquially known as the "Wasserman-Wolf problem")
Apparently the problem ought to be solved with a closed-form expression. See this paper for details.