I have a differential for a ray of light entering a medium with refractive index n(y), and using Fermat's principle for the path of light - the equation is given as:
$$ \frac{dy}{dx}=\sqrt {\frac{\left(1+10^{-4}e^{\left(-\frac{y}{5}\right)}\right)^2}{cos^2(\theta_0)}-1}$$
Where y is the height traveled, and $\theta_0$ is the angle above the x-axis.
I need to find the largest angle $\theta_0$ in which the graph of the function is no longer bounded, i.e. the graph doesn't approach a constant value.
At first I thought it had to do with small angle approximations, where if we take a value of $\theta_0$ that can no longer be approximated to 1,but that doesn't seem to be correct.
My next train of thought was seeing what happens when y approaches infinity, and then finding a condition on $\theta_0$, but that hasn't proven to be fruitful since the resulting function isn't bounded either.
i'd appreciate some insight into the problem