Implicit differential of diameter

Question

In my physics course there is a problem where the volume "V" of a sphere is filled with a gas. The sphere is released in a liquid, therefore the amount of gas in this volume "V" decreases because of concentration differences. If you evaluate a time dependent mass-balance the following diff. equation is to be solved for the diameter "D(t)", where $\alpha$ is a physical constant. Boundary conditions are: $D(t=0)=D_{0}$

$$ \frac{d}{dt}D^3(t)=(-24 \pi\alpha) D(t) $$

I would like to have some help evaluating the $\frac{d}{dt}D^3(t)$ term...

Answer 1

We can simplify the above equation as: $$\frac{\mathrm{d}(D(t))^3}{\mathrm{d}t}=3(D(t))^2\frac{\mathrm{d}D(t)}{\mathrm{d}t}=-24\pi\alpha D(t)$$ Using the chain rule of differentiation. Now let's simplify it a little more so that it looks like a neat Differential equation: $$D(t)\;{\mathrm{d}D(t)}=-8\pi\alpha \mathrm{d}t$$ This should be simple to integrate with the boundary conditions given. Can you proceed from here?

Answer 2

You need to use the chain rule. The easiest way to remember it is to "expand the fraction": $$\frac{\mathrm{d}D^3}{\mathrm{d}t}=\frac{\mathrm{d}D^3}{\mathrm{d}D}\frac{\mathrm{d}D}{\mathrm{d}t}=3D^2\frac{\mathrm{d}D}{\mathrm{d}t}$$