= Solution
For positive differentiable speed $c(y)$ and a smooth ray written as a graph $y(x)$, the <Fermat principle> minimizes travel time
$$
\mathcal T[y]=\int_{x_0}^{x_1}\frac{\sqrt{1+y'^2}}{c(y)}\,dx.
$$
A variation vanishing at the endpoints gives the <Euler-Lagrange equation> for $F=\sqrt{1+y'^2}/c$. Its derivatives are $F_y=-c'\sqrt{1+y'^2}/c^2$ and $F_{y'}=y'/(c\sqrt{1+y'^2})$. Thus
$$
\frac d{dx}F_{y'}-F_y=\frac{y''}{c(1+y'^2)^{3/2}}+\frac{c'}{c^2\sqrt{1+y'^2}}=0,
$$
and multiplying by $c^2(1+y'^2)^{3/2}$ gives
$$
\boxed{c(y)y''+c'(y)(1+y'^2)=0.}
$$
The displayed differential equation requires differentiability, beyond mere continuity of $c$. For constant positive speed it reduces to $y''=0$, so rays are straight lines; a straight segment minimizes travel time because it has the shortest Euclidean length.
For $c(y)=y$ in the positive-height region, the <Beltrami identity> gives $F-y'F_{y'}=1/[y\sqrt{1+y'^2}]=1/R$. Hence $y^2(1+y'^2)=R^2$, whose nonvertical solutions are $(x-h)^2+y^2=R^2$. Formally imposing the given boundary points fixes $h=1/2$, $R=1/2$, and gives the <ray in a linear-speed medium>
$$
\boxed{y(x)=\sqrt{x(1-x)},\qquad 0<x<1.}
$$
It is the upper semicircle of diameter one, with vertical limiting tangents at its endpoints.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/ib/paper-1-optical-ray.png]
{title=Formal semicircular ray with zero-speed ideal endpoints in the linear-speed medium}
There is a necessary endpoint qualification. The speed is zero at the two stated endpoints. For any rectifiable path entering positive height, $ds/y\geq |dy|/y$, and its travel time diverges logarithmically on leaving or approaching height zero. Thus \b[the semicircle is the formal interior ray, but there is no finite-time minimizer between the zero-speed endpoints]. For endpoints at height $\epsilon>0$ the circular ray has radius $\sqrt{1/4+\epsilon^2}$ and finite travel time $2\operatorname{arsinh}(1/(2\epsilon))$, which tends to infinity as $\epsilon\downarrow0$. This explains the intended semicircle as a limiting <hyperbolic geodesic>, without asserting finite travel time where $c=0$.
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