# Solving a specific system of differential equations for the Wilberforce Pendulum

I am investigating the motion of the Wilberforce pendulum and need to find a equation for position and angle over time.

I have found equations for the acceleration of the pendulum from the forces, and now have equations of the form:

$$m\frac{d^2z}{dt^2}=kz + Ɛθ$$ $$I\frac{d^2θ}{dt^2}=Jθ + Ɛz$$

Here, M, I, k, J, and Ɛ are constant.

Now, I need to solve these, but don't quite know how to. I have dealt with solving DEs, but not simultaneously. I have done some reading, and I have heard of using the Laplace transform, but I sadly don't know how to use that method, and the online resources that I have looked at have been not very helpful.

If anyone knows of any methods of solving these types of problems to find position and angle over time then I would really appreciate either dropping the name of the method and a place that I could learn about how to use it, or even some working for this particular problem.

Some specifics: I don't want to assume that the frequencies match, and want prefer to have most, if not all initial conditions as variables I can set, such as mass, inertia, initial displacement/angle, etc.

I suppose $$m,k,\varepsilon,I,J$$ are constant, i.e. do not depend on the time $$t$$? If so you can do the following:
1. Rewrite your system into the form $$\frac{d x}{dt} = A\cdot x$$ with $$x(t)\in \mathbb{R}^4$$ and $$A\in\mathbb{R}^{4\times 4}$$ a Matrix. Hence you rewrite your system into one, in which only first and zero order differential terms appear. An Example how this can be done is found here How to express a 2nd order ODE as 1st order ODE's?