Is there an analytical solution to the nonlinear ODE $\frac{dx}{d\theta} = -\sqrt{\frac{x^2}{4\cos^2\theta} - \cos^2\theta}$ over $\theta \in [0, \pi/2]$ with initial condition $x(0) = 2$? Using the substitution $y = \sin\theta$, I was able to transform this into $\frac{dx}{dy} = -\sqrt{\frac{x^2-4(1-y^2)^2}{4(1-y^2)^2}},$ but couldn't get much further.
Context: On a typical globe, one can only see half of the earth at a time. Waldo Tobler's slides on "Unusual Map Projections" mention the following amusing solution to this problem: simply wrap the earth around the globe twice (see image).
In trying to find an aesthetically pleasing conformal version of this construction by changing the shape of the globe, I found myself needing to solve the above ODE to determine its cross section; specifically, mapping latitude $\theta$ to distance from the polar axis, $x(\theta)$.
It can be solved numerically, of course, and that's what I did. I'm just curious about whether an analytical solution exists. The numerical solution looks a lot like $x(\theta) = 2\sqrt{\cos\theta}$, but isn't exactly the same.
In case you're curious, here is a render of the final result of my hacking on all this.