1. Homework Homework Assignment #19 Read Section 9.3 Page 521, Exercises: 1 – 41(EOO) Quiz next time Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

2. Homework, Page 521 Which of the following differential equations are first order? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Selections a, c, d, and f are first order differential equations.

3. Homework, Page 521 Verify that each given function is a solution of the differential equation. 5. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

4. Homework, Page 521 Verify that the given function is a solution of the differential equation. 9. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

5. Homework, Page 521 Solve, using separation of variables. 13. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

6. Homework, Page 521 Solve, using separation of variables. 17. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

7. Homework, Page 521 Solve, using separation of variables. 21. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

8. Homework, Page 521 Solve, using separation of variables. 25. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

9. Homework, Page 521 Solve the initial value problem. 29. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

10. Homework, Page 521 Solve the initial value problem. 33. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

11. Homework, Page 521 Solve the initial value problem. 37. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

12. Homework, Page 521 37. Continued. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

13. Homework, Page 521 Solve the initial value problem. 41. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

14. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Jon Rogawski Calculus, ET First Edition Chapter 9: Introduction to Differential Equations Section 9.3: Graphical and Numerical Models

15. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Most differential equations cannot be solved explicitly, but there are graphical and numerical methods that afford estimates sufficiently accurate for most requirements.

16. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

17. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

18. Definition – Isocline An isocline is the set of points on the x-y coordinate plane where the slope defined by the differential equation has a constant value. The next slide illustrates the process of drawing a slope field using isoclines. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

19. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

20. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

21. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The above slope field shows the rate of warming or cooling for an object placed into an environment at 40°F.

22. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Most differential equations have a uniqueness property, that is, there is a unique solution to the differential equation meeting a specific initial value criterion. The graph below is that of an exception to this property.

23. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Given a differential equation and an initial condition, we may use Euler’s method to approximate the function’s value at a nearby value of t using the formula:

24. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

25. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

26. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Using a Computer Algebra System or CAS, it is easy to rapidly evaluate Euler’s Method results for large numbers of very small increments. As shown on an earlier slide, the more numerous and smaller the intervals, the more accurate the result. The table below shows the results for a CAS evaluation of an Euler’s solution.

27. Use Euler’s Method with h = 0.1 to approximate the given value of y(t). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

28. Homework Homework Assignment #20 Review Section 9.3 Page 537, Exercises: 1 – 17 (EOO) Quiz next time Rogawski Calculus Copyright © 2008 W. H. Freeman and Company