Download Applications of Differential Equations

Glacier National Park, Montana
Photo by Vickie Kelly, 2004
Exponential Growth
and Decay
Greg Kelly, Hanford High School, Richland, Washington
Ex. Find the equation of the curve in the xy-plane that
passes through the given point and whose tangent at
a given point (x, y) has the given slope
y2
slope =
3 x
Point: (1, 1)
The number of bighorn sheep in a population increases at
a rate that is proportional to the number of rabbits present
(at least for awhile.)
So does any population of living creatures. Other things
that increase or decrease at a rate proportional to the
amount present include radioactive material and money in
an interest-bearing account.
If the rate of change is proportional to the amount present,
the change can be modeled by:
dy
 ky
dt

dy
 ky
dt
1
dy  k dt
y
1
 y dy   k dt
Rate of change is proportional
to the amount present.
Divide both sides by y.
Integrate both sides.
ln y  kt  C

1
 y dy   k dt
Integrate both sides.
ln y  kt  C
ln y
e
kt C
e
y  e e
C
kt
Exponentiate both sides.
When multiplying like bases, add
exponents. So added exponents
can be written as multiplication.

ln y
e
kt C
e
y  e e
C
kt
Exponentiate both sides.
When multiplying like bases, add
exponents. So added exponents
can be written as multiplication.
y  e e
C kt
y  Ae
kt
Since
 eC
is a constant, let  e
C
A.

y  e e
C kt
y  Ae
kt
y0  Ae
Since
 eC
is a constant, let  e
C
A.
1
k 0
At
t  0 , y  y0
.
y0  A
y  y0 e
kt
This is the solution to our original initial
value problem.

Exponential Change:
y  y0 e
kt
If the constant k is positive then the equation
represents growth. If k is negative then the equation
represents decay.

Population Growth
In the spring, a bee population will grow according to an
exponential model. Suppose that the rate of growth of the
population is 30% per month.
a) Write a differential equation to model the population
growth of the bees.
b) If the population starts in January with 20,000 bees, use
your model to predict the population on June 1st.
