If we have a general second-order equation of the form
we can reformulate this as a system of first-order equations. If we let
, then the above equation can be written as
which is a system of first-order equations.
Therefore we can reduce any second-order ODE to a system of first-order ODEs. Furthermore, using this approach we can reduce any higher-order ODE to a system of first-order ODEs. For example, a fourth-order ODE would yield a system of four first-order ODEs.
Therefore we only need to consider numerical techniques for solving first-order systems of ODEs, since any higher-order equation can simply be reduced to a system of first-order equations.