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Chapter  2 First-Order Chapter  2 First-Order

Chapter 2 First-Order - PowerPoint Presentation

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Uploaded On 2019-12-28

Chapter 2 First-Order - PPT Presentation

Chapter 2 FirstOrder Differential Equations 2 3 FIGURE 211 A solution curve is tangent to lineal element at 2 3 4 FIGURE 212 Solution curves following flow of a direction field 5 FIGURE 213 ID: 771620

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Chapter 2 First-Order Differential Equations

2

3FIGURE 2.1.1 A solution curve is tangent to lineal element at (2, 3)

4FIGURE 2.1.2 Solution curves following flow of a direction field

5FIGURE 2.1.3 Direction field and solution curves in Example 1

6FIGURE 2.1.4 Direction field in Example 2 on page 38

7FIGURE 2.1.5 Phase portrait of the DE in Example 3

8FIGURE 2.1.6 Lines y ( x ) = c 1 and y ( x ) = c 2 partition R into three horizontal subregions

9FIGURE 2.1.7 Phase portrait and solution curves in Example 4

10FIGURE 2.1.8 Behavior of solutions near y = 1 in Example 5

11FIGURE 2.1.9 Critical point c is an attractor in (a), a repeller in (b), and semi-stable in (c) and (d).

12FIGURE 2.1.10 Direction field for an autonomous DE

13FIGURE 2.1.11 Translated solution curves of an autonomous DE

14FIGURE 2.1.12 Direction field for Problem 1

15FIGURE 2.1.13 Direction field for Problem 2

16FIGURE 2.1.14 Direction field for Problem 3

17FIGURE 2.1.15 Direction field for Problem 4

18FIGURE 2.1.16 Graph for Problem 13

19FIGURE 2.1.17 Graph for Problem 14

20FIGURE 2.1.18 Graph for Problem 29

21FIGURE 2.1.19 Graph for Problem 30

22FIGURE 2.2.1 Solution curve for the IVP in Example 2

23FIGURE 2.2.2 Level curves of G ( x , y ) = e y + ye − y + e −y + 2 cos x

24FIGURE 2.2.3 Level curves c = 2 and c = 4

25FIGURE 2.2.4 Piecewise-defined solutions of (9)

26FIGURE 2.2.5 Shape of a cable in Problem 57

27FIGURE 2.2.6 Level curves in Problem 60

28FIGURE 2.3.1 Solution curves of DE in Example 2

29FIGURE 2.3.2 Solution curves of DE in Example 5

30FIGURE 2.3.3 Discontinuous f ( x ) in Example 6

31FIGURE 2.3.4 Graph of ( 10) in Example 6

32FIGURE 2.3.5 Graph of (15) in Example 7

33FIGURE 2.4.1 Solution curves of DE in Example 3

34FIGURE 2.4.2 Uncoiling chain in Problem 45

35FIGURE 2.5.1 Solutions of DE in Example 3

36FIGURE 2.6.1 Magnification of a neighborhood about the point (2, 4)

37FIGURE 2.6.2 Approximating y ( x 1) using a tangent line

38TABLE 2.6.1 h = 0.1

39TABLE 2.6.2 h = 0.05

40TABLE 2.6.3 h = 0.1

41TABLE 2.6.4 h = 0.05

42FIGURE 2.6.3 Comparison of the Runge-Kutta (RK4) and Euler methods

43FIGURE 2.6.4 A not-very helpful numerical solution curve

44FIGURE 2.R.1 Graph for Problem 13

45FIGURE 2.R.2 Graph for Problem 14

46FIGURE 2.R.3 Graph for Problem 16

47FIGURE 2.R.4 Portion of a direction field for Problem 17

48FIGURE 2.R.5 Graph for Problem 37

49FIGURE 2.R.6 Portion of a direction field for Problem 39

50FIGURE 2.R.7 Portion of a direction field for Problem 40