Exponential Growth & Differential Equations Solved by
Separation of
Variables
If y is a quantity whose rate of change (with respect to
time) is proportional to the quantity present at time t,
then y = Cekt where C is the
initial
value and k is the constant of proportionality.
PROOF:
-
dy = ky
dt
- dy = kdt (Separate
the variables)
y
-
(find
the integral of both sides)
- ln y = kt + C (the C
on the left side was transposed to the right and combined with the
constant there)
- y = e kt + C
(solving
for y)
- = eCe kt
(Separate the exponent into two)
- = Ae kt
(eC is a constant, so just write it as A)
Example:
In 1970 the population of a town was 3000
and
in 1980 it was 4000. Determine the
population of the town in 1990 if the growth rate is
proportional to the existing population.
-
dP = kP
dt
- dP = k dt (Separate
the variables)
P
-
(find the integral of both sides, using the limits for the
population
and time set by the problem)
-
1980
4000
ln P
|
= kt |
1970
3000
- ln 1980 - ln 1970 = 4000k - 3000k
-
ln 1980 = 1000k
1970
- k = 1
ln 1980
1000
1970
-
1990
N
ln P
|
= kt
|
(from above, using limits needed to answer the question)
1980
4000
-
1990
N
ln P
|
= 1 ln 1980
t | (Substituting
in the proper value for k)
1980
1000 1970
4000
- ln 1990 - ln 1980 =
1 ln
1980
(N - 4000)
1000 1970
- 1000 ln 1990
=
ln
1980 N - 4000 ln
1980
1980
1970
1970
- 1000 ln 1990 + 4000 ln 1980
= N ln
1980
1980
1970
1970
- 1000 ln 1990 + 4000 ln 1980
= N
1980
1970
ln 1980
1970
- 1000 ln 1990
1980 + 4000 = N
ln 1980
1970
- N = 4994.962
Newton's Law of Cooling:
The rate at which an object's temperature is changing at any given time
is
proportional to the difference between its temperature and the
temperature
of the surrounding medium.
dT = -k (T - To)
dt
dT
=
-kt
(Separate the variables by dividing by expression in parentheses)
T - To
ln |T - To | = -kt + C
(Integrate both sides)
|T - To | = e -kt + C
(Change to an exponential)
T - To = eC e -kt
(Rewrite the right side and remove the absolute value signs)
(The absolute value signs are absorbed into the constant on the
right.)
T = Ae -kt + To
(Add To)
Problems
Go to Differential
Equations and Mathematical Modeling
Go to Calculus
Lessons Page
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