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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.

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    If you look at the usual proof of existence in ODE textbooks for first-order nonlinear equations, there's an approach via fixed point of a mapping defined by taking an integral in the neighborhood of an initial point. To show the mapping is a contraction, and therefore has a fixed point, some strong continuity assumptions on the integrand are needed. In your case there's a singularity at t=0, so you would need to work "away" from that point.2011-08-24

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Let $y=a+X$ ,

Then $y'=X'$

$\therefore y'=\dfrac{t}{6}-\dfrac{y^2}{t^2}$

Let $y=\dfrac{t^2u'}{u}$ ,

Then $y'=\dfrac{t^2u''}{u}+\dfrac{2tu'}{u}-\dfrac{t^2(u')^2}{u^2}$

$\therefore\dfrac{t^2u''}{u}+\dfrac{2tu'}{u}-\dfrac{t^2(u')^2}{u^2}=\dfrac{t}{6}-\dfrac{t^2(u')^2}{u^2}$

$\dfrac{t^2u''}{u}+\dfrac{2tu'}{u}-\dfrac{t}{6}=0$

$6tu''+12u'-u=0$