Compute explicitly a plane curve $\displaystyle c(s) $ which is parameterized by arc length and whose curvature is $\displaystyle \kappa(s) = s^{-\frac{1}{2}} $.
we have $\displaystyle c(s)=(x(s), y(s)),$ where $\displaystyle x'(s)=\cos \theta, \ y'(s)=\sin \theta,$ where $\displaystyle \frac{d \theta}{d s} = \kappa = \frac{1}{\sqrt{s}}.$ thus one solution would be $\displaystyle \theta = 2\sqrt{s}$ and so $\displaystyle x'(s)=\cos(2\sqrt{s}), \ y'(s)=\sin(2\sqrt{s}).$
now integrate (substitute $\displaystyle 2\sqrt{s}=t$ and then use by parts) to find $\displaystyle x(s)$ and $\displaystyle y(s).$