Ordinary Differential Equations

CONTENTS ......Page 4
Preface for the Teacher ......Page 13
1. How Differential Equations Originate ......Page 14
1. Basic Concepts ......Page 15
2. The Meaning of the Terms Set and Function ......Page 19
3. The Differential Equation ......Page 34
4. The General Solution of a Differential Equation ......Page 42
5. Direction Field ......Page 52
2. Special Types of Differential Equations \r\nof the First Order ......Page 60
6. Meaning of the Differential of a Function ......Page 61
7. First Order Differential Equation with Homogeneous Coefficients ......Page 71
8. Differential Equations with Linear Coefficients ......Page 76
9. Exact Differential Equations ......Page 84
10. Recognizable Exact Differential Equations ......Page 94
11. The Linear Differential Equation of the First Order ......Page 105
12. Miscellaneous Methods of Solving a First Order Differential Equation ......Page 113
13. Geometric Problems ......Page 121
14. Trajectories ......Page 129
15. Dilution and Accretion Problems etc. ......Page 136
16. Motion of a Particle Along a Straight Line ......Page 152
17. Pursuit Curves. Relative Pursuit Curves ......Page 182
17M. Miscellaneous Types of Problems Leading to Equations of the First Order ......Page 197
4. Linear Differential Equations \r\nof Order Greater Than One ......Page 210
18. Complex Numbers and Complex Functions ......Page 211
19. Linear Independence of Functions ......Page 219
20. Solution of the Homogeneous Linear Differential Equation of order n with Constant Coefficients ......Page 225
21. Solution of the Nonhomogeneous Linear Differential Differential Equation of Order n with Constant Coefficients ......Page 235
22. Solution of the Nonhomogeneous Linear Differential Equation by the Method of Variation of Parameters ......Page 247
23. Solution of the Linear Differential Equation with Nonconstant Coefficients ......Page 255
5. Operators and Laplace Transforms ......Page 264
24. Differential and Polynomial Operators ......Page 265
25. Inverse Operators ......Page 282
26. Solution of a Linear Differential Equation by Means of the Partial Fraction Expansion of Inverse Operators ......Page 297
27. The Laplace Transform. Gamma Function ......Page 306
28. Undamped Motion ......Page 327
29. Damped Motion ......Page 361
30. Electric Circuits. Analog Computation ......Page 383
30M. Miscellaneous Types of Problems Leading to Linear Equations of the Second Order ......Page 394
31. Solution of a System of Differential Equations ......Page 407
32. Linearization of First Order Systems ......Page 438
33. Mechanical, Biological, Electrical Problems Giving Rise to Systems of Equations ......Page 454
34. Plane Motions Giving Rise to Systems of Equations ......Page 473
35. Special Types of Second Order Linear and Nonlinear Differential Equations Solvable by Reduction to a System of Two First Order Equations ......Page 514
36. Problems Giving Rise to Special Types of Second Order Nonlinear Equations ......Page 520
37. Power Series Solutions of Linear Differential Equations ......Page 545
38. Series Solution of y' = f(x,y) ......Page 562
39. Series Solution of a Nonlinear Differential Equation of Order Greater Than One and of a System of First Order Differential Equations ......Page 569
40. Ordinary Points and Singularities of a Linear Differential Equation. Method of Frobenius ......Page 584
41. The Legendre Differential Equation ......Page 605
42. The Bessel Differential Equation ......Page 623
43. The Laguerre Differential Equation ......Page 638
10. Numerical Methods ......Page 645
44. Starting Method. Polygonal Approximation ......Page 646
45. An Improvement of the Polygonal Starting Method ......Page 655
46. Starting Method—Taylor Series ......Page 659
47. Starting Method—Runge-Kutta Formulas ......Page 667
48. Finite Differences. Interpolation ......Page 673
49. Newton's Interpolation Formulas ......Page 677
50. Approximation Formulas Including Simpson's and Weddle's Rule ......Page 686
51. Milne's Method of Finding an Approximate Numerical Solution of y' = f(x,y) ......Page 698
52. General Comments. Selecting h. Reducing h. Summary and an Example ......Page 704
53. Numerical Methods Applied to a System of Two First Order Equations ......Page 716
54. Numerical Solution of a Second Order Differential Equation ......Page 721
55. Perturbation Method. First Order Equation ......Page 727
56. Perturbation Method. Second Order Equation ......Page 729
11. Existence and Uniqueness Theorem \r\nfor the First Order Differential \r\nEquation ......Page 733
57. Picard's Method of Successive Approximations ......Page 734
58. An Existence and Uniqueness Theorem for the First Order Differential Equation y' = f(x,y) ......Page 742
59. The Ordinary and Singular Points of a First Order Differential Equation y' = f(x,y) ......Page 758
60. Envelopes ......Page 761
61. The Clairaut Equation ......Page 771
62. An Existence and Uniqueness Theorem for a System of n First Order Differential Equations and for a Nonlinear Differential Equation of Order Greater Than One ......Page 777
63. Determinants. Wronskians ......Page 784
64. Theorems About Wronskians and the Linear Independence of a Set of Solutions of a Homogeneous Linear Differential Equation ......Page 792
65. Existence and Uniqueness Theorem for the Linear Differential Equation of Order n ......Page 797
Bibliography ......Page 805
Index ......Page 806