Even before taking into account the effect of air resistance, the velocity of a falling hailstone can be substantially slowed by other factors. Consider, for instance, a spherical hailstone falling. through a uniform mist under the influence of gravity alone. As the hailstone falls through the fog, it collects moisture to itself from the mist, gaining mass in a manner that affects its velocity. The hailstone's vertical displacement y is then governed by a non-linear second-order differential equation:
d²y/dt² = -g - 3/y(dy/dt)²
Use appropriate substitution methods to determine a general solution for v(y), the vertical velocity v as a function of vertical displacement y.