I am working on a Physics problem. The problem involves a ball of mass m falling through the air, with the drag force at a time t being equal to $bv^3(t)$. The problem wants me to find and solve a differential equation for the instantaneous velocity at a given time. I did some research, and I found an article on the Oregon State University website which explains how to solve the equation assuming that drag is directly proportional to velocity. I tried solving it the same way, and I eventually end up at a point where an auxiliary function $p'(t) = -2\frac{b^2}{m^2}p^2(t)$. This is pretty much where the problem dead-ended for me, though.
My two questions are:
1) Is there an elementary function / combination of elementary functions such that $f'(t)\propto f^2(t)$?
2) How do I actually solve this equation for $v(t)$?
1) Is there an elementary function / combination of elementary functions such that $f′(t)∝f^2(t)$ ?
Yes. Take for example $f(x)=1/x$ we have $f'(x)=-\frac 1 {x^2}$ and also $f^2(x)=\frac 1 {x^2}$ $$p'(t) = -2\frac{b^2}{m^2}p^2(t)$$ This differential equation is separable. If you solve the equation in p you get : $$\int \frac {dp}{p^2}= -2\int \frac{b^2}{m^2}dt$$
$$\frac 1 p=2\frac {b^2}{m^2}t+K$$ Invert fractions on both sides you get $$ p(t)=\frac 1{ {2\frac {b^2}{m^2}t+K}}$$ multiply denominator and numerator by $m^2$ $$ p(t)=\frac {m^2} { {2 {b^2}t+Km^2}}$$ Substitute $C=m^2K$ which is just a constant
$$\implies p(t)=\frac {m^2} {2{b^2}t+C}$$