1. First-Order Differential Equations
Chapter 2
First-Order Differential Equations

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

4. FIGURE 2.1.2 Solution curves following flow of a direction field

5. FIGURE 2.1.3 Direction field and solution curves in Example 1

6. FIGURE 2.1.4 Direction field in Example 2 on page 38

7. FIGURE 2.1.5 Phase portrait of the DE in Example 3

8. FIGURE 2.1.6 Lines y(x) = c1 and y(x) = c2 partition R into three horizontal subregions

9. FIGURE 2.1.7 Phase portrait and solution curves in Example 4

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

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

12. FIGURE 2.1.10 Direction field for an autonomous DE

13. FIGURE 2.1.11 Translated solution curves of an autonomous DE

14. FIGURE 2.1.12 Direction field for Problem 1

15. FIGURE 2.1.13 Direction field for Problem 2

16. FIGURE 2.1.14 Direction field for Problem 3

17. FIGURE 2.1.15 Direction field for Problem 4

18. FIGURE 2.1.16 Graph for Problem 13

19. FIGURE 2.1.17 Graph for Problem 14

20. FIGURE 2.1.18 Graph for Problem 29

21. FIGURE 2.1.19 Graph for Problem 30

22. FIGURE 2.2.1 Solution curve for the IVP in Example 2

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

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

25. FIGURE 2.2.4 Piecewise-defined solutions of (9)

26. FIGURE 2.2.5 Shape of a cable in Problem 57

27. FIGURE 2.2.6 Level curves in Problem 60

28. FIGURE 2.3.1 Solution curves of DE in Example 2

29. FIGURE 2.3.2 Solution curves of DE in Example 5

30. FIGURE 2.3.3 Discontinuous f(x) in Example 6

31. FIGURE 2.3.4 Graph of (10) in Example 6

32. FIGURE 2.3.5 Graph of (15) in Example 7

33. FIGURE 2.4.1 Solution curves of DE in Example 3

34. FIGURE 2.4.2 Uncoiling chain in Problem 45

35. FIGURE 2.5.1 Solutions of DE in Example 3

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

37. FIGURE 2.6.2 Approximating y(x1) using a tangent line

38. TABLE h = 0.1

39. TABLE h = 0.05

40. TABLE h = 0.1

41. TABLE h = 0.05

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

43. FIGURE 2.6.4 A not-very helpful numerical solution curve

44. FIGURE 2.R.1 Graph for Problem 13

45. FIGURE 2.R.2 Graph for Problem 14

46. FIGURE 2.R.3 Graph for Problem 16

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

48. FIGURE 2.R.5 Graph for Problem 37

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

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