DIFFERENTIAL EQUATIONS
7.5 Exponential Growth and Decay
Objective: Use differential equations to solve growth and decay problems
I. A population grows at a rate proportional to the size of the population
II. A substance decays at a rate proportional to the size of the mass
III. The value of a savings account at continuously compounded interest increases at a rate
proportional to that value
IV. Law of natural growth: k is a positive constant
dy ky
dt =
V. Law of natural decay: k is a negative constant
dy ky
dt =
VI. Solve the initial-value problem y(0) = y 0
dy ky
dt =
A. dy = (ky)dt dy kdt
y
⇒=
B. C. dy kdt
y=
∫∫
C. ln |y| = kt + C |y| =
⇒
ktCCkt
eee
+=
D. where A is an arbitrary constant
kt
yAe=c
(eor0)=±
E. kt
0
y(t)ye=
VII. The relative growth rate k is constant
A. If then
dP kP,
dt =1dP
kPdt
=
B. If and t is in years, then k = .02
dP .02P,
dt =
1. Population grows at a rate of 2% per year
2. where P0 is the initial population
.02t
0
P(t)Pe=
VIII. Example 2 on p. 530
A. The initial-value problem is P(0) = 2520
dP kP,
dt =
B. The solution is c
kt
P(t)2520e=
