1. 2/18/2016Calculus - Santowski1 C.3.2 - Separable Equations Calculus - Santowski

2. 2/18/2016Calculus - Santowski2 Lesson Objectives 1. Solve separable differential equations with and without initial conditions 2. Solve problems involving exponential decay in a variety of application (Radioactivity, Air resistance is proportional to velocity, Continuously compounding interest, Population growth)

3. 2/18/2016Calculus - Santowski3 (A) Separable Equations So far, we have seen differential equations that can be solved by integration since our functions were relatively easy functions in one variable Ex. dy/dx = sinx - 1/x Ex. dv/dt = -9.8

4. 2/18/2016Calculus - Santowski4 (A) Separable Equations In Ex 1, we simply evaluated the indefinite integral of both sides But what about the equation dy/dx = -x/y? If we tried finding antiderivatives or indefinite integrals ….

5. 2/18/2016Calculus - Santowski5 (A) Separable Equations If we have equations in the form of where, then such that Then we can rewrite the equation as and integrate as before

6. 2/18/2016Calculus - Santowski6 (B) Examples Given the DE (a) Solve (b) Graph

7. 2/18/2016Calculus - Santowski7 (B) Examples Given the DE (a) Solve (b) Graph

8. 2/18/2016Calculus - Santowski8 (B) Examples Given the DE (a) Solve (b) Graph

9. 2/18/2016Calculus - Santowski9 (B) Examples Given the DE (a) Solve (b) Graph

10. 2/18/2016Calculus - Santowski10 (B) Examples Given the DE (a) Solve given (b) Graph the solutions on a slope field diagram

11. 2/18/2016Calculus - Santowski11 (B) Example Here is the graphic solution for

12. 2/18/2016Calculus - Santowski12 (C) - Application - Exponential Growth Write a DE for the statement: the rate of growth of a population is directly proportional to the population Solve this DE

13. 2/18/2016Calculus - Santowski13 (C) - Application - Exponential Growth Write a DE for the statement: the rate of growth of a population is directly proportional to the population Solve this DE:

14. 2/18/2016Calculus - Santowski14 (D) Examples The population of bacteria grown in a culture follows the Law of Natural Growth with a growth rate of 15% per hour. There are 10,000 bacteria after the first hour. (a) Write an equation for P(t) (b) How many bacteria will there be in 4 hours? (c) when will the number of bacteria be 250,000?

15. 2/18/2016Calculus - Santowski15 (D) Examples The concentration of phosphate pollutants in a lake follows the Law of Natural Growth with a decay rate of 5.75% per year. The phosphate pollutant concentrations are 125 ppm in the second year. (a) Write an equation for P(t) (b) What will there be phosphate pollutant concentration in 10 years? (c) A given species of fish can be re-introduced into the lake when the phosphate concentration falls below 35 ppm. When can the fish be re-introduced?

16. 2/18/2016Calculus - Santowski16 (E) Homework See handouts (1) Salas, §9.2, p457, Q1-20 (2) Dunkley, §7.5, p371, Q1-7