1. SPECIALIST MATHS Differential Equations Week 1

2. Differential Equations The solution to a differential equations is a function that obeys it. Types of equations we will study are of the form:

3. Obtaining Differential Equations To obtain a differential equation from a function, we must: differentiate the function, then manipulate the result to achieve the appropriate equation.

4. Example 1 (Ex 8B1) Show that is a solution of the differential equation

5. Solution 1 Show that is a solution of the differential equation

6. Example 2 (Ex 8B1) Show that is a solution of the differential equation

7. Solution 2 Show that is a solution of the differential equation Solution

8. Example 3 (Ex 8B1) Show that is a solution of the differential equation

9. Solution 3 Show that is a solution of the differential equation Solution Now

10. Example 4 (Ex 8B2) Given is the solution of the differential equation Find a, b, c and d given

11. Solution 4 Given is the solution of the differential equation Find a, b, c and d given Solution:

12. Solution 4 continued

13. Example 5 (Ex 8B2) Find a, b, c, and d if is the solution of and

14. Solution 5 Find a, b, c, and d if is the solution of and Solution:

15. Solution 5 continued

16. Solution 5 continued again

17. Slope Fields The differential equation gives a formula for the slope its solutions. For example the differential equation gives an equation to calculate the slopes of all points in the plane for functions whose derivatives are. That is it gives the slopes of all points of functions of the form

18. Slope Field for f ‘(x) = 2x x=0 x=1x=2 x=-1 x=-2 y x

19. Slope Field Generator y’ = 2x for y = x 2 + c y‘ = 3x 2 for y = x 3 + c y’ = 2x + 1 for y = x 2 + x + c y’ = x y’ = y y’ = x + y http://alamos.math.arizona.edu/ODEApplet/JOdeApplet.html

20. Example 6 (Ex 8C1) Solve the following differential equation

21. Solution 6 Solve the following differential equation Solution:

22. Example 7 (Ex 8C1) Solve

23. Solution 7 Solve Solution

24. Solution 7 continued

25. Solution 7 continued again

26. Euler’s Method of Numerical Integration We find the solution of a differential equation by moving small increments along the slope field Start at (x o,y o ), then move up the slope field and at the same time going out horizontally h to get to the next point (x 1,y 1 ). The smaller the value of h the more accurate the solution.

27. Euler’s Method

28. Fundamental Theorem of Calculus Using Euler’s method if we make the size of h very small then the y value of the point we approach is given by:

29. Example 8 (Ex 8C2) Use Euler’s method with 3 steps to find y(0.6) for the differential equation with y(0)=2 Find y(6) using the Fundamental theorem

30. Solution 8 Use Euler’s method with 3 steps to find y(0.6) for the differential equation with y(0)=2 Find y(6) using the Fundamental theorem Solution:

31. Solution 8 continued

32. Solution 8 continued again

33. This week Exercise 8A1 Q2, 3 Exercise 8B1 Q 1 – 7 Exercise 8B2 Q 1 – 7 Exercise 8C1 Q 1 – 7 Exercise 8C2 Q 1, 2