Elementary Differential Equations Fifth EditionPDF电子书下载

Chapter 1．Definitions and Elementary Problems 1

1．General remarks 1

2．Differential equation．Order．Degree 2

3．Solution of a differential equation 3

4．Geometric considerations 5

5．Existence theorems 7

6．General solution．Particular solution 8

7．Finding differential equation from general solution 8

8．Variables separable 11

9．Review exercises 13

Chapter 2．Applications 15

10．Geometric applications using rectangular coordinates 15

11．Orthogonal trajectories 16

12．Geometric applications using polar coordinates 18

13．Use of limits 20

14．Physical applications 21

15．Compound-interest-law problems 22

16．Acceleration．Velocity．Distance 24

17．Other rate problems 27

18．Miscellaneous problems 30

Chapter 3．Differential Equations of the First Order and the First Degree 33

19．Simple substitutions 33

20．Homogeneous equations 35

21．Equations of the form(ax+by+c)dx+(αx+βy+γ)dy=0 37

22．Exact differentials 38

23．Exact differential equations 39

24．Integrating factors 42

25．Linear differential equation 45

26．Equations reducible to linear form 48

27．Simultaneous equations 49

28．Summary 51

Chapter 4．Applications Involving Differential Equations of the First Order 54

29．Miscellaneous elementary applications 54

30．Applications involving simultaneous equations 56

31．Applications to the flow of electricity 59

32．Air pressure 60

33．Applications involving forces and velocities 61

34．Review problems 67

Chapter 5．First-order Equations of Degree Higher than the First 70

35．Foreword 70

36．Equations solvable for dy/dx 70

37．Envelopes 72

38．Envelope from differential equation 75

39．Equations solvable for y 78

40．Equations solvable for x 80

41．Review problems 81

Chapter 6．Linear Differential Equations with Constant Coefficients 83

42．Operators 83

43．Linear independence of functions 86

44．Linear differential equation 88

45．Homogeneous linear differential equation with constant coefficients 89

46．Auxiliary equation has repeated roots 90

47．Constants of integration from initial conditions 91

48．Auxiliary equation has imaginary roots 92

49．Right-hand member not zero 94

50．Special case when the right-hand member is not zero 96

51．A basic theorem relating to operators 97

52．Methods using symbolic operators 98

53．Variation of parameters 103

54．Simultaneous differential equations 106

55．Summary and review exercises 108

Chapter 7．Laplace Transforms 110

56．Introduction 110

57．Definition of a Laplace transform 110

58．Some properties of Laplace transforms 111

59．Deriving transform relations from given ones 114

60．Inverse transforms of products 117

61．Transforms of derivatives 121

62．Solving differential equations by transforms 122

63．Solving systems of differential equations 124

64．Resolving a fraction into partial fractions 126

65．Fractions having repeated factors in the denominator 128

66．Partial fractions．Quadratic factors 130

67．Review problems 133

Chapter 8．Applications of Linear Equations with Constant Coefficients 135

68．Harmonic motion．Damping 135

69．Types of damping．Resonance 137

70．Forces．Accelerations．Moments 139

71．Some fundamental equations of motion 140

72．Oscillatory motion 141

73．Plane motions of bodies 147

74．Kirchhoff＇s current law and electromotive-force law 151

75．Simple circuits containing constant electromotive force 153

76．Simple circuits containing a sinusoidal electromotive force 154

77．Resonance 155

78．Applications of Kirchhoff＇s laws to networks 156

79．Review problems 161

Chapter 9．Miscellaneous Differential Equations of Order Higher than the First 163

80．Reduction of order by substitution 163

81．Dependent variable absent 163

82．Independent variable absent 165

83．Method based on factorization of the operator 166

84．Euler＇s linear equation 169

85．Second-order linear equation 170

86．Review exercises 171

Chapter 10．Applications 173

87．Radius of curvature 173

88．Cables．The catenary 174

89．Equation of elastic curve．Beams 176

90．Columns 181

91．Motion of a particle in a plane 182

92．Review problems 185

Chapter 11．Existence Theorems and Applications 187

93．Foreword 187

94．Replacement of differential equations by a system of the first order and first degree 187

95．Existence theorems 188

96．Differential equations of the first order and first degree in the unknowns 191

97．Total differential equations 194

98．Geometrical interpretation 198

99．Fields of force in space 198

100．Review exercises 200

Chapter 12．Solution by Series 202

101．Introduction 202

102．Integration in series 204

103．Solution involving a more general type of series 208

104．Indicial equation has roots differing by an integer 210

105．The gamma function 213

106．Bessel＇s equation 215

107．Bessel＇s functions 217

108．Expansion of functions in terms of Bessel＇s functions 222

109．Legendre＇s functions 224

Chapter 13．Numerical Solutions of Differential Equations 229

110．Introduction 229

111．Methods of successive approximations 229

112．Newton＇s interpolation formula 231

113．Interpolation 234

114．Formulas for approximate integration 236

115．Illustration and discussion of formulas(16)to(21)，§114 237

116．Halving the interval h of x 240

117．Numerical solution of a system of simultaneous equations 242

118．The Runge-Kutta method 245

Chapter 14．Partial Differential Equations 248

119．Introduction 248

120．Solution of a partial differential equation 248

121．Equations easily integrable 250

122．Equations having the form Pp+Qq=R 251

123．Finding particular solutions satisfying given conditions 254

124．Separation of variables 256

125．Hyperbolic，parabolic，elliptic equations 258

Chapter 15．Applications of Partial Differential Equations 263

126．Fourier series 263

127．Cosine series．Sine series 267

128．Application to nuclear fission 269

129．Vibrations of a string 272

130．Vibrations of a rod 275

131．Flow of heat 276

132．One-dimensional heat flow 279

133．Vibrations of a membrane 281

134．Telephone，telegraph，and radio equations 283

135．Fluid motion 286

Answers 291

Index 313