ordinary differential equations - Calculating roller coaster loops - how to get x and y in terms of s?

In this video, the author presents a method to calculate shapes of roller coaster loops. At 13:20, three differential equations are presented to plot the shape of a loop providing a constant force $G$ for an initial velocity $v_0$:

$\begin{align}
\frac{\partial\theta}{\partial s} &= \frac{G-g\cos\left(\theta\right)}{v_0^2-2gy} \\
\frac{\partial x}{\partial s} &= \cos\theta \\
\frac{\partial y}{\partial s} &= \sin\theta
\end{align}$

where $g$ is acceleration due to gravity, $9.80665~\text{m}/\text{s}^2$.

I would like to eliminate the need for the $\frac{\partial\theta}{\partial s}$ term. Do I integrate both sides of all these equations with respect to $s$ and then substitute $\theta$ in for the $x$ and $y$ equations, or am I stuck with three equations?

Another problem is that the first equation has $\theta$ and $y$ on the right side, so how would I proceed?