1. Solving Differential Equations Slope Fields

2. Solving DE: Slope Fields Slope Fields allow you to approximate the solutions to differential equations graphically.

3. Solving DE: Slope Fields Draw a slope field in the grid for the given differential equation. Then, draw in a solution curve for the initial condition. Example 1 Initial condition

4. Solving DE: Slope Fields Draw a slope field in the grid for the given differential equation. Then, draw in a solution curve for the initial condition. Example 1 Initial condition

5. Solving DE: Slope Fields Draw a slope field in the grid for the given differential equation. Then, draw in a solution curve for the initial condition. Example 1 Initial condition

6. Solving DE: Slope Fields Draw a slope field in the grid for the given differential equation. Then, draw in solution curves for the initial conditions. Example 2 Initial condition

7. Solving DE: Slope Fields Draw a slope field in the grid for the given differential equation. Then, draw in solution curves for the initial conditions. Example 2 Initial condition

8. Solving DE: Slope Fields Draw a slope field in the grid for the given differential equation. Then, draw in solution curves for the initial conditions. Example 2 Initial condition

9. Solving DE: Slope Fields Draw a slope field in the grid for the given differential equation. Then, draw in solution curves for the initial conditions. Example 2 Initial condition

10. Solving DE: Slope Fields Draw a slope field in the grid for the given differential equation. Then, draw in solution curves for the initial conditions. Example 3 Initial condition

11. Solving DE: Slope Fields Draw a slope field in the grid for the given differential equation. Then, draw in solution curves for the initial conditions. Example 3 Initial condition

12. Solving DE: Slope Fields Draw a slope field in the grid for the given differential equation. Then, draw in solution curves for the initial conditions. Example 3 Initial condition

13. Solving DE: Slope Fields Draw a slope field in the grid for the given differential equation. Then, draw in solution curves for the initial conditions. Example 3 Initial condition

14. Solving DE: Slope Fields Assignment  WS Slope Fields  Due Friday, 04 December 2015.  WS Solving Differential Equations – Slope Fields  Due Friday, 04 December 2015.

15. Solving Differential Equations Euler’s Method (pronounced “Oiler”)

16. Solving DE: Euler’s Method Illustration of Euler’s Method; the curve is being “linearized” to approximate a desired solution, given an initial value. A better approximation is attained with a smaller step size h.

17. Solving DE: Euler’s Method Euler’s Method To approximate the solution of the initial-value problem y ′ = f (x, y), y (x 0 ) = y 0, proceed as follows: Step 1:Choose a nonzero number h to serve as an increment or step size along the x-axis, and let x 1 = x 0 + h, x 2 = x 1 + h, x 3 = x 2 + h,... NOTE: h will be given.

18. Solving DE: Euler’s Method Euler’s Method Step 2:Compute successively y 1 = y 0 + f (x 0, y 0 )h y 2 = y 1 + f (x 1, y 1 )h y 3 = y 2 + f (x 2, y 2 )h... The numbers y 1, y 2, y 3,... in these equations are the approximation of y(x 1 ), y(x 2 ), y(x 3 ),...

19. Solving DE: Euler’s Method Ex:Use Euler's method with a step size of 0.5 to solve the initial-value problem y' = y – x, y (0) = 2 over the interval 0 ≤ x ≤ 1.

20. Solving DE: Euler’s Method Ex:Use Euler's method with a step size of 0.5 to solve the initial-value problem y' = y – x, y (0) = 2 over the interval 0 ≤ x ≤ 1.

21. Solving DE: Euler’s Method Ex:Use Euler's method with a step size of 0.25 to solve the initial-value problem y' = y – x, y (0) = 2 over the interval 0 ≤ x ≤ 1.

22. Solving DE: Euler’s Method Ex:Use Euler's method with a step size of 0.5 to solve the initial-value problem y' = y – x, y (0) = 2 over the interval 0 ≤ x ≤ 1.

23. Solving DE: Euler’s Method Ex:Use Euler's method with a step size of 0.5 to solve the initial-value problem y' = y – x, y (0) = 2 over the interval 0 ≤ x ≤ 1. When using Euler’s Method, use all decimals !

24. Solving DE: Euler’s Method WS Euler’s Method  Due on Tuesday, 08 December 2015. Looking Ahead  Monday, 07 December 2015: Review  Tueday, 08 December 2015: TEST Indefinite Integration  Wed/Thur, 09/10 December 2015: TEKS Exam  Friday, 11 December 2015: Final Exam – Part I (calculator)  Monday, 14 December 2015: Final Exam – Part II (no calculator)

25. AP Calculus BC Friday, 04 December 2015 OBJECTIVE TSW solve differential equations using the technique of separation of variables. ASSIGNMENTS DUE –Sec. 5.6: pp. 408-409 (17-23 odd, 24, 26, 33-35 all, 37)  to the left of the wire basket –WS Graphs of Derivatives  wire basket –WS Slope Fields  black basket –WS Solving Differential Equations – Slope Fields  to the right of the black tray TODAY’S ASSIGNMENT –WS Solving Differential Equations – Separation of Variables  due on Tuesday, 08 December 2015

26. WS: Euler’s Method

27. Solving Differential Equations Separation of Variables

28. Solving Differential Equations: Separation of Variables We have seen differential equations that are explicitly in terms of x:

29. Solving Differential Equations: Separation of Variables We solved these by multiplying both sides by dx and integrating. For example,

30. Solving Differential Equations: Separation of Variables Sometimes, though, the differential equations are not as straightforward. This is called the general solution because it has "C". NOTE: The solution is NOT Get terms with "y" on one side, all other terms on the other side.

31. Solving Differential Equations: Separation of Variables Let's add an initial condition: y(0) = 1 This is called the particular solution.

32. Solving Differential Equations: Separation of Variables Solve The equation cannot be simplified for y.

33. Solving Differential Equations: Separation of Variables For the particular solution, y(0) = 0, so

34. Solving Differential Equations: Separation of Variables WS Separation of Variables  Due on Tuesday, 08 December 2015 (TEST day).

35. WS: Separation of Variables