In mathematics and numerical computing, there is really a standard way to recondition any ODE of order N into a system:
If the unknown is really y (not y'), this ODE is equivalent to the system
u = y'
u' = u + 1
that is of order 1 with 2 unknown, and that requires a (y0, u0) couple of initial conditions.
And that's it.
IMO, this is definitely not a topic to work on.