In case you're curious about context---there isn't one. I am thinking about this out-of-the-blue because it is mysterious and interesting.
Let $ a, p, q, y_0 $ be positive constants. Consider the ordinary differential equation in $ y = y(x) $,
$$ y' = y^p - a x^q, y(0) = y_0$$
Let us only consider the behavior of the solution for $ x \geq 0 $ and $ y(x) \geq 0 $. Depending on the constants, the solution will either go to infinity, or go to zero (we consider the solution undefined after $ y(x) < 0 $). This is obvious on inspection, but in case it's not obvious to you, I recommend you plot some numerical solutions for $ p = 2, q = 3, a = 1 $. 
$ p = 2, q = 3, a = 1 $">
I am investigating the conditions under which this solution escapes to infinity. I am not aware of any closed-form solutions for this ODE. I'd like to understand the boundary between solutions that are escaping to infinity and those that are going to zero (it's not hard to show that there's no other possibility).
I don't know of many tools here. From my undergrad ODEs, I am vaguely familiar with a Lyapunov method. Viz., we might try to define a function $ F = F(x,y) $ satisfying some inequality like $$ (*) \frac{d}{dx} \left[ F(x, y(x)) \right] \geq 0 $$
Such functions, if chosen well, may allow us to put certain upper envelopes on solutions that go to zero, and put lower envelopes on solutions that go to infinity. In the following discussion, always assume $ x \geq 0, y \geq 0 $.
For instance, if we try $ F(x,y) = y^{r_1}(y^{r_2} + b x^{r_3}) $ for some appropriate (positive) constants $ b, r_1, r_2, r_3 $, we have $ F = 0 $ only if $ y = 0 $. If we can additionally show that $ (*) $ holds, then we produce a lower-envelope for solutions that escape to infinity. There is some hope that this might work. Indeed, we have $$ \frac{d}{dx}\left[ F(x, y(x)) \right] = \frac{\partial F}{\partial x} + \frac{\partial F}{\partial y} y' = \frac{\partial F}{\partial x} + \frac{\partial F}{\partial y} (y^p - a x^q) $$ It seems like some wizardry with the arithmetic-geometric inequality and good choice of constants can allow one to make this non-negative. Can anyone help?