A pendulum with a horizontally oscillating fulcrum can be described using the Lagrange formalism (related to a setting, as it is shown e.g. here) by the following system of nonlinear ordinary differential equations (ODE):
$$\begin{cases} m\ddot x+kx+ml\left(\ddot \theta \cos \left(\theta\right)-\dot \theta^2 \sin \left(\theta\right)\right) & \text{= 0} \\ \cos \left(\theta\right) \ddot x+g\sin \left(\theta\right)+l \ddot \theta & \text{= 0} \end{cases},$$
with $m$ being the pendulum's point mass, $l$ being the length of the pendulum's massless, inextensible and always taut cord, $k$ being the spring constant, $g \approx 9.81 \frac{m}{s^2}$ being Earth's average gravitational acceleration, $x$ being the horizontal deflection of the spring, and $\theta$ being the angle of rotation of the pendulum.
After I had comprehended the derivation, I asked myself how the equations of motion would look like in concrete terms. Of course, it's hardly possible to find analytical solutions for these equations.
But what is the easiest way to generate numerical approximations? Is it possible online, with appropriate programs (e.g. Scilab; if so, what would be the code?) or at all? Does the system have to be somehow manipulated, linearized, simplified ... before (if so, how?)? Which and how many boundary conditions are necessary?
Thank you and all the best!