For a sphere of radius r the volume is V=4/3*Pi*r^3 and the surface area is A=4*Pi*r^2

Write and solve an ODE for r(t) under the following assumptions

a) The rate of change of the radius is proportional to the surface area.

b) The rate of change of the surface area is proportional to the radius.

c) The rate of change of the volume is inversely proportional to the radius.

d) The rate of change of the volume is inversely proportional to the surface area.

My solution:

a)dr/dt=k*A, where k>0 then dr/dt=4*k*Pi*r^2

b) dA/dt=k*r, k>0

But dA/dt=dA/dr * dr/dt then dA/dr * dr/dt = k*r then dr/dt = k/8*Pi

c) dV/dt=k/r, k>0

But dV/dt = dV/dr * dr/dt = 4*Pi*r^2 * (dr/dt)

Then dr/dt = k/(4*Pi*r^3)

d) dV/dt = k/A

But dV/dt = dV/dr * dr/dt = 4*Pi*r^2 * dr/dt

Then dr/dt = k/(16*Pi*r^4)

My question is that for part a,c,d I end up with a non-linear differential equation which I don't know how to solve since I am a first year.

I suspect that something went completely wrong with my calculations, any help will be hugely appreciated. Thanks in advance!