I'm working on this homework problem and running into an issue. The problem is 2.4.2 from A First Course in Dynamics by Hasselblatt & Katok. Consider the differential equation $\dot{x} = - x^k$, where $k > 1$, and let $x_0 > 0$ be an initial condition. The problem says to show that there exists $s > 0$ such that $\lim_{t \to \infty} x(t) / t^s$ is finite and nonzero.
I solved the separable ODE as follows: \begin{align*} \frac{\mathrm{d} x}{\mathrm{d} t} & = - x^k \\ \Rightarrow - x^{-k} \mathrm{d} x & = \mathrm{d} t \\ \Rightarrow \int - x^{-k} \mathrm{d} x & = \int \mathrm{d} t \\ \Rightarrow - \frac{x^{-k + 1}}{-k + 1} & = t + C \\ \Rightarrow x^{-k + 1} & = (k - 1)t + C \\ \Rightarrow x & = \left[ (k - 1) t + C \right]^{\frac{1}{- k + 1}} \\ & = \left[ (k - 1) t + C \right]^{\frac{- 1}{k -1}} . \end{align*} From what I can tell, this means that $x(t) = O \left( t^{-1/(k - 1)} \right)$, so if $s > 0$, then $x(t) / t^s = O \left( t^{- \frac{1}{k - 1} - s} \right)$, meaning the limit as $t \to \infty$ will be $0$.
Is there an error in this problem? Is it suppose to be that $\lim_{t \to \infty} x(t) \cdot t^s$ is finite and nonzero, where $s = \frac{1}{k - 1}$? Or am I missing something?
Thanks in advance for your help!