Autonomous ordinary differential equation

The following equation has appeared in this forum several times:

$y d^2 y/d t^2 = a (dy /dt)^2$

for instance,

solving ODE with power law solution

The same equation, but in a form corresponding to an intermediate step towards its full solution,

$d y/d t = c y^a$

is also thoroughly discussed in several books on ordinary differential equations, for example,

Handbook of Ordinary Differential Equations: Exact Solutions, Methods, and Problems, Polyanin, Andrei D and Zaitsev, Valentin F, 2017.

However, and surprisingly, I haven’t seen any application of this equation to the modelling of any physical/economical/social system. Most likely, I’m simply not using the right expressions when looking for it.

Does someone know of any references where this equation arises or is derived in the modelling of a real system?