1. 6.4 Applications of Differential Equations

2. I. Exponential Growth and Decay
A.) Law of Exponential Change - Any situation where a quantity (y) whose rate of growth or decay at any time t is directly proportional to the amount of the quantity present at that time t.
B.) Examples- Population growth, Radioactive decay, Compounding continuously

3. II. Procedure 1.) Set up the function.
2.) Solve the differential equation.
3.) Find all relevant constants. (Use Table)
4.) Solve the problem.
5.) Graph the equations.

4. III. General Case
Exponential Growth/Decay Model -

5. IV. Newton’s Law of Cooling
Any situation where the rate of change of an object’s temperature (T) is proportional to the difference between its temperature T and the temperature TS of the surrounding medium, assuming TS stays fairly constant.

6. V. Examples
See Handout

7. 1.) According to U.N. data, the world population in the beginning of 1975 was 4 billion and growing at a rate of 2% per year. Assuming an exponential growth model, estimate the population in the year Start from your initial differential equation with this problem and use calculus to solve it.

8. Solution

9. 2. ) A ½ - life of a certain radioactive element is 25 years
2.) A ½ - life of a certain radioactive element is 25 years. How much of 1 gram remains after 15 years? You may use the general case exponential function to solve this problem.

10. Solution

11. 3.) A hard-boiled egg at 98ºC is put in a pan under running water that is 18 ºC. After 5 minutes, the egg is found to be 38 ºC. How much longer will it take the egg to reach 20 ºC? Start from your initial differential equation with this problem and use calculus to solve it.

12. Solution