Elastic pendulum problem solver

This webpage uses the Runge-Kutta-Fehlberg fourth-order method with fifth-order error checking (RKF45) to approximate the solution to the problem of the elastic pendulum: \[ \begin{aligned} \dfrac{d^{2}x}{dt^2} &= (l_0 + x) \dot{\theta}^2 - \dfrac{kx}{m} + g \sin{\theta} \\ \dfrac{d^{2} \theta}{dt^2} &= -\dfrac{g}{l_0 + x} \cos{\theta} - \dfrac{2\dot{x}\dot{\theta}}{l_0 + x} \end{aligned} \]

Parameter	Value	Explanation
\(g\):		Problem parameter.
\(l_0\):		Problem parameter.
\(k\):		Problem parameter.
\(m\):		Problem parameter.
\(t_0\):		Starting time for the simulation in seconds (s).
\(t_f\):		End time for the simulation in seconds.
\(x_0\):		Value of \(x\) at \(t_0\).
\(\dot{x}_0\):		Value of \(\dot{x}\) at \(t_0\).
\(\theta_0\):		Value of \(\theta\) at \(t_0\).
\(\dot{\theta}_0\):		Value of \(\dot{\theta}\) at \(t_0\).
\(\epsilon\):		Error tolerance. \(\epsilon \lt\) 1.8e-10 usually freezes the webpage up.
dtInitial:		Initial guess for step size.