We now apply the method of separation of variables to three classes of functions
associated with different rates of change. In each of the three cases, we describe
the rate of change of a quantity, write the differential equation that follows from
the description, then solve—or, in some cases, just give the solution of—the d.e.
We list several applications of each case, and present relevant problems
involving some of the applications.
Case I: Exponential Growth
An interesting special differential equation with wide applications is defined by
the following statement: “A positive quantity y increases (or decreases) at a rate
that at any time t is proportional to the amount present.” It follows that the
quantity y satisfies the d.e.