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: \[ \begin{aligned} \dfrac{d^{2}x}{dt^2} &= (l_0 + x) \dot{\theta}^2 - \dfrac{kx}{m} + g \sin{\theta} \\ \dfrac{d^{2} \theta}{dt^2} &= -\dfrac{g}{l_0 + x} \cos{\theta} - \dfrac{2\dot{x}\dot{\theta}}{l_0 + x} \end{aligned} \]