Modify the equations (Example 2 on page 314 of Apostol) for the velocity of a falling body in a resisting medium if the resistance of the medium is proportional to instead of to . Prove that the resulting differential equation can be written in each of the following forms:
where . By integrating these find the following formulas for :
where . Determine the value of as .
Starting with the equation
in example 2 and modifying it so that the resistance is proportional to we have
Using the chain rule as we did in the previous exercise we know
Letting we then have
This is the first requested equation.
Integrating the first equation we find,
Integrating the second equation,