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# Use the chain rule to rewrite and solve a given differential equation

(This exercise is in reference to Example 2 on page 314 of Apostol, though I’ve added some details to hopefully make it self-contained.)

The problem of finding the speed of a falling body of mass in a resisting medium with resistance under the influence of the Earth’s gravitational attraction is governed by the differential equation

If is the distance the body has fallen at time then . Use the chain rule to write

Use this to show that the first-order differential equation for a falling body in a resisting medium can be rewritten as

where

Express in terms of by integrating this differential equation and check this result against the results derived in Example 2 which tell us that

Considering as a function of and applying the chain rule, we have

Thus,

Then, integrating we have

for some constant . Since when we have that . Furthermore we know,

Therefore,

This is the result we expected from Example 2.