I am looking for explanations how we could classify the curve (spiral) defined by differential equation in polar coordinates: $r'^2+r^2=(kt)^2$, $r(t=0)=0$, k- Const.
For small angles the curve is similar to Galileo spiral which then transforms into Archimedean spiral: a) if $\phi \in (0,t)$, t is quite small, then $r(\phi) \approx k/2 *\phi^2$ b) if $\phi \in (c, + \infty)$, c is quite large positive number, then $r(\phi)=k\phi+o(\phi)$, as $\phi \to \infty$.
I thought that this is a 'mechanical' curve. Indeed, according to OEIS the junction point of the curve and the ray uniformly rotated in the origin coordinates moves uniformly accelerated. However I failed to find any references to be sure that is a mechanical curve. The graphic of the curve is also available to see.
So what's the right way to classify the curve?
Any explanations are highly welcomed.
PS I think the curve is not an algebraic one as well.