Well... in process I found that
1) I have missed one negative sign in the Newton's law: m * Y'' = - c * (Y')^2 - m*g
2) I need to include the Magnus force to account for an extra force that can change the trajectory of the tire due to pressure difference on different sides of the rotating tire (a variant of Bernoulli effect).
3) The gyroscopic effect, that is the conservation of the angular momentum of a rotating tire, has to be accounted somehow because the Magnus force may exert torque on the rotating tire.
4) I read several relatively new academic papers on kinematics of tennis balls and frisbees.
... S$!t , this is an active research field! We need more data, I guess ... or not?