Reducing higher order ODEs

If we have a general second-order equation of the form a(x)d2ydx2+b(x)dydx+c(x)y=f(x), we can reformulate this as a system of first-order equations. If we let z=dydx, then the above equation can be written as dydx=z, dzdx=f(x)b(x)zc(x)ya(x), 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.