Asymptotic expansion of a simple ODE 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 $\epsilon &gt; 0$, 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 $q$?
hint: Consider two options for larger $t$:

$|x|^p&lt;&lt;t^{-q}$

$|x|^p &gt;&gt; t^{-q}$


solve for each case and check for consistency.
hint: Consider two options for larger $t$:

$|x|^p&lt;&lt;t^{-q}$

$|x|^p &gt;&gt; t^{-q}$


solve for each case and check for consistency.