In order to to calculate desired light path in continuous medium with gradient refraction index, for schematic see the Figure below.
$O:(0,0)$ is the disk center of light source $\odot{O}$ with radius $3$.
Then the profile light rays of disk $O$ from the view point $B:(-14,0)$ is defined by segments $DB$ and $EB$ (also the tangent lines of $\odot{O}$ through $B$) when the refraction index is a constant value everywhere.
Now if the refraction index is defined as:
$$n(x,y)=\dfrac{e^{\tfrac{(x+15)^2+y^2+12}{(x+15)^2+y^2+11}}}{e}$$
How to determine the two profile light curves of disk $O$ from the viewpoint $B$?
I tried to establish:
$$\delta\int{n(x,y)}\rm{d}s=0$$ and the second order nonlinear ODE via Euler-Lagrange equation: $$y''(x)=\dfrac{2\left((x+15)y'(x)-y(x)\right)\left(y'(x)^2+1\right)}{\left(y(x)^2+x(x+30)+236\right)^2}$$ but don't know how to establish initial/boundary values of the ordinary differential equation and obtain a symbolic or numerical solution.
Update
Since $y(-14)=0$ is easily available, actually my question is only `how to determine another' $y'(-14)=?$ such that the ODE can be easily solved numerically?
Update
I tried some calculation, and it seems, for any numerical solution $y(x)$, it will be difficult to determine whether it is tangent to the circle $O$ or not:
