Can I get a symbolic solution for these coupled ODEs?

I found this IPython notebook called ‘Roller Coaster’ from", "numfys.net, where they model the movement of a ball over a path described by a third-degree polynomial $$y(x)$$ with slope $$\theta(t)$$ using the following following coupled set of ordinary differential equations (ODE)

$$\frac{\text dv}{\text dt} = \frac{g\sin (\theta(t))}{1 + I_0/mr^2},$$

where $$v$$ is the momentary velocity along the track. The $$x$$ position is in turn given by

$$\text dx = \text ds \cos(\theta)\: \Longrightarrow \: \frac{\text dx}{\text dt} = v\cos(\theta).$$

Figure of a ball rolling on curve $$𝑦(𝑥)$$. The forces acting on the object are indicated in the figure. At each point $$𝑥$$, the slope of the curve is defined by an angle $$𝜃(𝑥)$$.

How can I obtain a symbolic solution for this set of ODE?

• It seems that you have 2 equations and 3 unknowns, that might not be enough to solve the problem. – nicoguaro May 7 at 14:14