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

Therefore,

Letting we then have

This is the first requested equation.

Alternatively,

Integrating the first equation we find,

Integrating the second equation,

This implies

The starting condition is v(0)=0, so there is no typo in the book, but there is in the proposed solution for the first integral here.

The constant of the first integral is \frac{m}{k} \log c. The constant of the second integral is 0.

RoRi, I noticed that you approached the first integration (that of \frac{ds}{dv}) the same way that I did. Namely, working with the assumption that Tom made a typo in the formula for

The way that you and I solved the problem, the formula is as such

But the book has it written as such, so that the factor \frac{gm}{k} distributes.

It threw me for a loop when I solved the equation. I thought it might have been a clever way to get rid of the constant of integration, but I couldn’t figure out a way to get it to distribute the way it was written in the book