Differential Equations Review
May 02, 2010
Exponential Growth: A positive quantity y increases or decreases at a
rate that at any time t is proportional to the amount present.
where k > 0 if y is increasing and k < 0 if y is decreasing, then solving
the linear differential equation for y, we get:
Ms. Waldron
Differential Equations Review
Ex 1: The population of a country is growing at a rate proportional to
its population. If the growth rate per year is 4% of the current
population, how long will it take for the population to double?
Ms. Waldron
May 02, 2010
Differential Equations Review
May 02, 2010
Ex 2: The bacteria in a certain culture increases continuously at a rate
proportional to the number present.
a) If the number triples in 6 hours, how many will be there in 12 hours?
b) In how many hours will the original number quadruple?
Ms. Waldron
Differential Equations Review
May 02, 2010
Ex 3: Radium-226 decays at a rate proportional to the quantity present.
Its half-life is 1612 years. How long will it take for one quarter of a given
quantity of radium 226 to decay?
If Q(t) is the amount present at time t, then it satisfies the equation
Ms. Waldron
Differential Equations Review
May 02, 2010
In 1970 the world population was approximately 3.5 billion. Since then it
has been growing at at rate proportional to the population, and the factor of
proportionality has been 1.9%/year. At that rate, in how many years would
there be one person per square foot of land (The land area of Earth is
2
15 2
approximately 200,000,000 mi , or about 5.5 x 10 ft )
Ms. Waldron
Differential Equations Review
May 02, 2010
Ex 5: Because of limited food and space, a squirrel population cannot
exceed 1000. It grows at a rate proportional to the exiting population.
If there were 100 squirrels 2 years ago, and 1 year ago the population
was 400, how many squirrels are there now?
Ms. Waldron
Differential Equations Review
Ex 6: Newton's Law of Cooling
Ms. Waldron
May 02, 2010