The above 2nd order differential equation has straightforward solutions of:

A natural question is then: can we do something like this with higher order equations?

Gave the general solutions of the higher order polynomial equations:

Finally, he froze the screen on a set of fourth order differential equations.

The answer is yes for second order equations, but there's more work to do.

The original second order equation may then finally be integrated:

The integrating factor for a general second order differential equation:

It evolves in time according to the second order differential equation:

The researchers discovered that a second order mathematical equation would more precisely fit the data.

These are second order differential equations for the four metric functions.