Vertical motion of a spherical body with decreasing mass

The paper studies the fall and ascent in air of a spherical body with radius linearly decreasing with time. The first integral of the nonlinear equation of motion is expressed in closed form in terms of Bessel functions on the assumption that the aerodynamic drag depends on the velocity as a quadratic polynomial. The second integral is evaluated by an approximate formula. The reliability of the analytic solution is confirmed by comparing it with a numerical solution of the Cauchy problem