A particle falling under gravity is subjet to a retarding force proportional to its velocity. Find its position as a function of time, if it starts from rest, and show that it will eventually reach a terminal velocity.
2. The attempt at a solution
To save myself some latex time, I skipped a couple of the intermediary steps here:
integrating both sides
This final equation I get is correct (hopefully my procedure is too) and at this point, the question offers a hint:
[HINT: After the first integration, use an integrating factor, i.e. a function f(t) such when the equation is multiplied by f(t), the left-hand side becomes an exact derivative, in fact, the derivative of xf. The final stage requires an integration by parts]
This is where I got stuck. How do I determine the integrating factor f(t)? And what exactly is meant by "the left-hand side becomes an exact derivative, in fact, the derivative of xf"?
Your help is appreciated, thanks!