1. 1 Differential Equations 6 Copyright © Cengage Learning. All rights reserved. 6.1 DE & Slope Fields BC Day 1

2. 2 Differential Equations and Slope Fields Copyright © Cengage Learning. All rights reserved. 6.1

3. 3 Use initial conditions to find particular solutions of differential equations. Use slope fields to approximate solutions of differential equations. Use Euler’s Method to approximate solutions of differential equations. Objectives

4. 4 General and Particular Solutions

5. 5 A differential equation in x and y is an equation that involves x, y, and derivatives of y. A differential equation gives the slope of a solution curve at any point in the plane in terms o the coordinates of the point. A function y = f(x) is called a solution of a differential equation if the equation is satisfied when y and its derivatives are replaced by f(x) and its derivatives.

6. 6 General and Particular Solutions For example, differentiation and substitution would show that y = e –2x is a solution of the differential equation y' + 2y = 0. Show it. It can be shown that every solution of the above differential equation is of the form y = Ce –2x General solution of y ' + 2y = 0 where C is any real number. This solution is called the general solution.

7. 7 General and Particular Solutions The order of a differential equation is determined by the highest- order derivative in the equation. For instance, y' = 4y is a first-order differential equation. The second-order differential equation s''(t) = –32 has the general solution s(t) = –16t 2 + C 1 t + C 2 which contains two arbitrary constants. Show that s(t) is a general solution of the second order differential equation. It can be shown that a differential equation of order n has a general solution with n arbitrary constants.

8. 8 The term “initial condition” stems from the fact that, often in problems involving time, the value of the dependent variable or one of its derivatives is known at the initial time t = 0. For instance, the second-order differential equation s''(t) = –32 having the general solution s(t) = –16t 2 + C 1 t + C 2 General solution of s''(t) = –32 might have the following initial conditions. s(0) = 80, s'(0) = 64 Initial conditions In this case, the initial conditions yield the particular solution s(t) = –16t 2 + 64t + 80. Particular solution General and Particular Solutions

9. 9 Example 1 – Verifying Solutions Determine whether the function is a solution of the differential equation y '' – y = 0. a. y = sin x b. y = 4e –x c. y = Ce x Solution: a. Because y = sin x, y' = cos x, and y'' = –sin x, it follows that y'' – y = –sin x – sin x = –2sin x ≠ 0. So, y = sin x is not a solution.

10. 10 Example 1 – Solution b. Because y = 4e –x, y' = –4e –x, and y'' = 4e –x, it follows that y'' – y = 4e –x – 4e –x = 0. So, y = 4e –x is a solution. c. Because y = Ce x, y' = Ce x, and y'' = Ce x, it follows that y'' – y = Ce x – Ce x = 0. So, y = Ce x is a solution for any value of C. cont’d

11. 11 General and Particular Solutions Geometrically, the general solution of a first-order differential equation represents a family of curves known as solution curves, one for each value assigned to the arbitrary constant. For instance, you can verify that every function of the form is a solution of the differential equation xy' + y = 0. Show it.

12. 12 Figure 6.1 shows four of the solution curves corresponding to different values of C. Particular solutions of a differential equation are obtained from initial conditions that give the values of the dependent variable or one of its derivatives for particular values of the independent variable. Figure 6.1 General and Particular Solutions

13. 13 Example 2 – Finding a Particular Solution For the differential equation xy ' – 3y = 0, verify that y = Cx 3 is a solution, and find the particular solution determined by the initial condition y = 2 when x = –3. Solution: You know that y = Cx 3 is a solution because y' = 3Cx 2 and xy'– 3y = x(3Cx 2 ) – 3(Cx 3 ) = 0.

14. 14 Example 2 – Solution Furthermore, the initial condition y = 2 when x = –3 yields y = Cx 3 General solution 2 = C(–3) 3 Substitute initial condition. Solve for C Particular solution and you can conclude that the particular solution is Try checking this solution by substituting for y and y' in the original differential equation cont’d xy ' – 3y = 0

15. 15 Integrating to find solutions:  Use integration to find a general solution of the differential equation:

16. 16 You try  Use integration to find a general solution of the differential equation:

17. 17 6.1 Slope Fields Greg Kelly, Hanford High School, Richland, Washington

18. 18 A little review: Consider: then: or It doesn’t matter whether the constant was 3 or -5, since when we take the derivative the constant disappears. However, when we try to reverse the operation: Given: find We don’t know what the constant is, so we put “C” in the answer to remind us that there might have been a constant. This is the general solution

19. 19 If we have some more information we can find C. Given: and when, find the equation for. This is called an initial value problem. We need the initial values to find the constant. An equation containing a derivative is called a differential equation. It becomes an initial value problem when you are given the initial condition and asked to find the original equation.

20. 20 Slope Fields

21. 21 Slope Fields Solving a differential equation analytically can be difficult or even impossible. However, there is a graphical approach you can use to learn a lot about the solution of a differential equation. Consider a differential equation of the form y' = F(x, y) Differential equation where F(x, y) is some expression in x and y. At each point (x, y) in the xy–plane where F is defined, the differential equation determines the slope y' = F(x, y) of the solution at that point.

22. 22 Slope Fields If you draw short line segments with slope F(x, y) at selected points (x, y) in the domain of F, then these line segments form a slope field, or a direction field, for the differential equation y' = F(x, y). Each line segment has the same slope as the solution curve through that point. A slope field shows the general shape of all the solutions and can be helpful in getting a visual perspective of the directions of the solutions of a differential equation. Slope fields are graphical representations of a differential equation which give us an idea of the shape of the solution curves. The solution curves seem to lurk in the slope field.

23. 23 Slope Fields A slope field shows the general shape of all solutions of a differential equation.

24. 24 Sketching a Slope Field Sketch a slope field for the differential equation by sketching short segments of the derivative at several points.

25. 25 Draw a segment with slope of 2. Draw a segment with slope of 0. Draw a segment with slope of 4. 000 010 00 00 2 3 10 2 112 204 0 -2 0-4

26. 26 If you know an initial condition, such as (1,-2), you can sketch the curve. By following the slope field, you get a rough picture of what the curve looks like. In this case, it is a parabola. Slope fields show the general shape of all solutions of a differential equation. We can see that there are several different parabolas that we can sketch in the slope field with varying values of C.

27. 27 Slope Fields  Create the slope field for the differential equation Since dy/dx gives us the slope at any point, we just need to input the coordinate: At (-2, 2), dy/dx = -2/2 = -1 At (-2, 1), dy/dx = -2/1 = -2 At (-2, 0), dy/dx = -2/0 = undefined And so on…. This gives us an outline of a hyperbola

28. 28 Given: Let’s sketch the slope field … Slope Fields

29. 29 Separate the variables Given f(0)=3, find the particular solution.

30. 30 C Slope Fields In order to determine a slope field for a differential equation, we should consider the following: i) If points with the same slope are along horizontal lines, then DE depends only on y ii) Do you know a slope at a particular point? iii) If we have the same slope along vertical lines, then DE depends only on x iv) Is the slope field sinusoidal? v) What x and y values make the slope 0, 1, or undefined? vi) dy/dx = a( x ± y ) has similar slopes along a diagonal. vii) Can you solve the separable DE? 1. _____ 2. _____ 3. _____ 4. _____ 5. _____ 6. _____ 7. _____ 8. _____ Match the correct DE with its graph: AB C E G D F H H B F D G E A

31. 31 Hints on matching a differential equation with its slope field :  If is only in terms of x, slopes will be parallel along vertical lines.  If is only in terms of y, slopes will be parallel along horizontal lines.  Horizontal slopes occur where y' = 3y – 6x will have horizontal slopes where y=2x.  Very steep slopes often correspond to a denominator = 0.  will have zero slopes where y= -x and vertical slopes where y=x.  If y' is periodic in x, the slope field will appear periodic.  If slopes are all positive or negative in any quadrant, look for y' containing the product xy or a quotient of the two. y' = 0.

32. 32 Slope Fields  Which of the following graphs could be the graph of the solution of the differential equation whose slope field is shown?

33. 33  1998 AP Question: Determine the correct differential equation for the slope field: Slope Fields

34. 34 Example – Sketching another Slope Field Sketch a slope field for the differential equation y' = x – y for the points (–1, 1), (0, 1), and (1, 1). Solution: The slope of the solution curve at any point (x, y) is F (x, y) = x – y. So, the slope at (–1, 1) is y' = –1 –1 = –2, the slope at (0, 1) is y' = 0 – 1 = –1, and the slope at (1, 1) is y' = 1 – 1 = 0.

35. 35 Example 3 – Solution Draw short line segments at the three points with their respective slopes, as shown in Figure 6.2. Figure 6.2 cont’d

36. 36 BC Homework Day 1: Pg. 409: 13-15 odds, 19-23 odds,31, 37-47, 53-59, odds on all Day 2 (Euler’s Lesson) : Pg. 410: 69,71,73 and Slope Fields Worksheet

37. 37 Homework  Slope Fields Worksheet  BC add pg. 411 69-73 odd