How reduce the differential equation to first order ?
For example:
and write the resulting system of equations in the form:
It's not really a formula. It's a procedure. You let higher-order derivatives be components of the vector z. Here's another link.
Technically, your original DE is order 5. However, since you left out the y term (the one with no derivative), you can use the substitution HallsofIvy mentioned, and immediately get a 4th order DE in u. Then, once you've found u, integrate once more to get y.