3.2 Exponential Growth and Decay

Summary

Differential equations whose solutions involve exponential growth or decay are discussed. Everyday real-world problems involving these models are also introduced.

By the end of your studying, you should know:

How to write as a differential equation the fact that the rate of change of the size of a population is increasing (or decreasing) in proportion to the size.
How to solve the IVP dy/dt = ky, where y(0) is specified and k is a constant.
How to solve exponential growth and decay word problems.
The meaning of doubling time and half-life.
Newton's Law of Cooling.

On-screen applet instructions: The applet shows the graph of y = Be^kx. Use the vertical slider to change k, and the horizontal slider to change B.

Examples

We have a 100-gallon tank filled with water. Saltwater is being pumped into the tank through two pipes.

Write an equation which states how the amount of salt S is changing with respect to time.
Use the differential equation from part 1 to write a function that states how much salt is present in the tank at a given time t.

A cup of coffee is initially 170 degrees Fahrenheit and is left in a room with ambient temperature 70 degrees Fahrenheit. Suppose that when the coffee is first placed in the room, it is cooling at a rate of 20 degrees per minute. Assuming Newton's law of cooling applies, how long does it take for the coffee to cool to 110 degrees?

The population of fish in a pond is modeled by the differential equation

where time t is measured in years. Towards what number does the population of fish tend? If there are initially 10 fish in the pond, how long does it take for the number of fish to reach 90% of the eventual population?