A typical second order differential equation is that of a harmonic oscillator,
which has a rather simple analytic solution, but is not immediately solvable using our Runge-Kutta or Euler method.
To make this problem accessible for our numerical methods we start with the following definitions:
After substituting the appropriate terms into our original equation, we obtain the following coupled first order equations,
We thus solve a single second order differential equation by simultaneously solving two first order equations. In classical mechanics this corresponds to transforming the Newtonian form of the equations of motions into the Hamiltonian. In general all differential equations can be transformed into sets of first order differential equations.