7.4. SEPARABLE DIFFERENTIAL EQUATIONS
423
Finally, we need to solve for y. Here, one point deserves careful attention. By the definition
of the natural logarithm function, it follows that
|y| = e
3t+C = e
3t e
C .
Since C is an unknown constant, e C is as well, though we do know that it is positive
(because e x is positive for any x). When we remove the absolute value in order to solve for
y, however, this constant may be either positive or negative. We will denote this updated
constant (that accounts for a possible + or −) by C to obtain
y(t) = Ce
3t .
There is one more slightly technical point to make. Notice that y = 0 is an equilibrium
solution to this differential equation. In solving the equation above, we begin by dividing
both sides by y, which is not allowed if y = 0. To be perfectly careful, therefore, we will
typically consider the equilibrium solutions separably. In this case, notice that the final
form of our solution captures the equilibrium solution by allowing C = 0.
Activity 7.8.
Suppose that the population of a town is growing continuously at an annual rate of 3%
per year.
(a) Let P(t) be the population of the town in year t. Write a differential equation
that describes the annual growth rate.
(b) Find the solutions of this differential equation.
(c) If you know that the town’s population in year 0 is 10,000, find the population
P(t).
(d) How long does it take for the population to double? This time is called the
doubling time.
(e) Working more generally, find the doubling time if the annual growth rate is k
times the population.
⊳
Activity 7.9.
Suppose that a cup of coffee is initially at a temperature of 105 ◦ F and is placed in a
75 ◦ F room. Newton’s law of cooling says that
dT
dt
= −k(T − 75),
423
Finally, we need to solve for y. Here, one point deserves careful attention. By the definition
of the natural logarithm function, it follows that
|y| = e
3t+C = e
3t e
C .
Since C is an unknown constant, e C is as well, though we do know that it is positive
(because e x is positive for any x). When we remove the absolute value in order to solve for
y, however, this constant may be either positive or negative. We will denote this updated
constant (that accounts for a possible + or −) by C to obtain
y(t) = Ce
3t .
There is one more slightly technical point to make. Notice that y = 0 is an equilibrium
solution to this differential equation. In solving the equation above, we begin by dividing
both sides by y, which is not allowed if y = 0. To be perfectly careful, therefore, we will
typically consider the equilibrium solutions separably. In this case, notice that the final
form of our solution captures the equilibrium solution by allowing C = 0.
Activity 7.8.
Suppose that the population of a town is growing continuously at an annual rate of 3%
per year.
(a) Let P(t) be the population of the town in year t. Write a differential equation
that describes the annual growth rate.
(b) Find the solutions of this differential equation.
(c) If you know that the town’s population in year 0 is 10,000, find the population
P(t).
(d) How long does it take for the population to double? This time is called the
doubling time.
(e) Working more generally, find the doubling time if the annual growth rate is k
times the population.
⊳
Activity 7.9.
Suppose that a cup of coffee is initially at a temperature of 105 ◦ F and is placed in a
75 ◦ F room. Newton’s law of cooling says that
dT
dt
= −k(T − 75),