1. 6.2 Differential Equations: Growth and Decay (Part 1)
Glacier National Park, Montana
Photo by Vickie Kelly, 2004
Greg Kelly, Hanford High School, Richland, Washington

2. Objectives
Use separation of variables to solve a simple differential equation.
Use exponential functions to model growth and decay in applied problems.

3. You know how to solve
And
when the equations have already been separated.

4. Solve the differential equation
Solution set of all equations where

5. The number of bighorn sheep in a population increases at a rate that is proportional to the number of sheep 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:

6. Growth and Decay Models:
Rate of change is proportional to the amount present.
Divide both sides by y.
Integrate both sides.

7. Integrate both sides.
Exponentiate both sides.
When multiplying like bases, add exponents. So added exponents can be written as multiplication.

8. Exponentiate both sides.
When multiplying like bases, add exponents. So added exponents can be written as multiplication.
Since is a constant, let

9. Since is a constant, let
At ,
This is the solution to our original initial value problem.

10. Theorem 5.16: Exponential Growth and Decay
If y is a differentiable function of t such that y>0 and y'=ky for some constant k, then
Exponential Change:
If the constant k is positive then the equation represents growth. If k is negative then the equation represents decay.

11. Example: The rate of change of y is proportional to y. When t=0, y=2
Example: The rate of change of y is proportional to y. When t=0, y=2. When t=2, y=4. What is the value of y when t=3?

12. Homework
6.2 (page 420)
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