The differential equation in polar coordinates $r'^2+r^2=(kt)^2$, $r(t=0)=0$, k- Const, for large $t$ presents Archimedes’ Spiral and Galileo's spiral for $t \to 0$.

I find it surprisingly, however I failed to find the fact in literature (some more details available at Analytical solutions of a differential equation (from Archimedes' Spiral) ).

So, I wonder if you know where the first publication of the equation appeared and what the context of the research was? I guess it was related to optic research. Any reference and ideas are highly welcomed.

Thank you in advance.