Asymptotic expansion of a simple ODE

Question

I'm interested in the long-time behavior of the simple ODE

$$\dot{x} = -x^p + 1/t^q$$

with $p, q$ integers not necessarily equal and $x = x(t)$, $\dot{x} = \frac{d}{dt}x$. Furthermore, $x(0) \geq 0$ so we do not need to worry about instability with $x$ negative for $p$ even. To avoid the singularity at zero we can feel free to say $1/(t+\epsilon)^q$ for some $\epsilon > 0$, it does not matter to me. Is there a simple method to say that $x \sim 1/t^l$ for some $l$ in terms of $p$ and $q$?

Answer 1

hint: Consider two options for larger $t$:

1. $|x|^p<<t^{-q}$

2. $|x|^p >> t^{-q}$

solve for each case and check for consistency.