I'm curious about techniques for solving a nonautonomous* system in the case of a non-linear differential equation. There's a simple example in my textbook (Hirsch, Smale, Devaney) where we obtain the following nonautonomous equation (after linearizing about the origin)
$ x'(t) = x + y_0^2 e^{-2 t}$
In this case, we simply guess a particular solution (which is obvious from the given equation) and everything follows through in a straightforward manner. In a paper I'm working through I have an equation that looks something like
$X'(t) = \frac{t}{6} - \frac{(a + X)^2}{t^2},$
and I'm not sure what strategies I should have at my disposal here. The literature on non-autonomous non-linear systems seems to be rather scarce from the bit of googling I've done. Any insight would be much appreciated.
*By non-autonomous I simply mean there is a $t$ hanging around on the right-side of the equation.