When x = e^4 , we have thus ln^2 y = 9 and ln y = 3 (where we chose ln y > 0 because y
> 1), so y = e^3.EXAMPLE 11
Find the general solution of the differential equation
SOLUTION: We rewrite
Taking antiderivatives yields eu = ev + C, or u = ln(ev + c).E. EXPONENTIAL GROWTH AND DECAY
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.
where k > 0 if y is increasing and k < 0 if y is decreasing.
From (1) it follows that
Then
If we are given an initial amount y, say y 0 at time t = 0, then
y 0 = c · ek · 0 = c · 1 = c,and our law of exponential change
tells us that c is the initial amount of y (at time t = 0). If the quantity grows with time, then k > 0; if it
decays (or diminishes, or decomposes), then k < 0. Equation (2) is often referred to as the law of