Thread: ODEs

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!

You're correct with all your DE's except for d), where you need a instead of

You're correct in that all of your DE's in a), c), and d) are nonlinear. However, they are all separable. Surely you've studied how to solve separable DE's?

Thanks Ackbeet. I know how to solve separable equations, I'll try to solve them and post you any question. So is just solving them with A'level stuff (separation of variables, not any other way)?

Originally Posted by Darkprince

Thanks Ackbeet. I know how to solve separable equations, I'll try to solve them and post you any question. So is just solving them with A'level stuff (separation of variables, not any other way)?

Sounds good. Right. I don't think you need anything beyond separation of variables to finish the problems.

Hello Ackbeet just solved the equations, if you have some time to confirm. The initial condition was r(0) = R

a) r(t)= -R/(4*k*Pi*R*t - 1)

b) r(t)= (K/8*Pi)*t + R

c) r(t) = ((K*Pi)*t + R^4)^1/4

d) r(t) = ((5*k/16 * Pi^2) * t + R^5)^1/5

Appreciate your help and your time, thank you.

The cool thing about ODE's is that verifying your solution is quite straight-forward, really (since differentiation is so much easier than integration). Just plug your solution into the DE and see if you get an equality. And make sure to do the same thing with the initial conditions. So, what do you get when you do that?

I'm gonna check that and if I find any differences I will let you know. Thanks again!

You're welcome!