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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

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

                                              Ball rolling

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?

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

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