Elementary Differential Equations Fifth Edition2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

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Chapter 1．Definitions and Elementary Problems1

1．General remarks1

2．Differential equation．Order．Degree2

3．Solution of a differential equation3

4．Geometric considerations5

5．Existence theorems7

6．General solution．Particular solution8

7．Finding differential equation from general solution8

8．Variables separable11

9．Review exercises13

Chapter 2．Applications15

10．Geometric applications using rectangular coordinates15

11．Orthogonal trajectories16

12．Geometric applications using polar coordinates18

13．Use of limits20

14．Physical applications21

15．Compound-interest-law problems22

16．Acceleration．Velocity．Distance24

17．Other rate problems27

18．Miscellaneous problems30

Chapter 3．Differential Equations of the First Order and the First Degree33

19．Simple substitutions33

20．Homogeneous equations35

21．Equations of the form(ax+by+c)dx+(αx+βy+γ)dy=037

22．Exact differentials38

23．Exact differential equations39

24．Integrating factors42

25．Linear differential equation45

26．Equations reducible to linear form48

27．Simultaneous equations49

28．Summary51

Chapter 4．Applications Involving Differential Equations of the First Order54

29．Miscellaneous elementary applications54

30．Applications involving simultaneous equations56

31．Applications to the flow of electricity59

32．Air pressure60

33．Applications involving forces and velocities61

34．Review problems67

Chapter 5．First-order Equations of Degree Higher than the First70

35．Foreword70

36．Equations solvable for dy/dx70

37．Envelopes72

38．Envelope from differential equation75

39．Equations solvable for y78

40．Equations solvable for x80

41．Review problems81

Chapter 6．Linear Differential Equations with Constant Coefficients83

42．Operators83

43．Linear independence of functions86

44．Linear differential equation88

45．Homogeneous linear differential equation with constant coefficients89

46．Auxiliary equation has repeated roots90

47．Constants of integration from initial conditions91

48．Auxiliary equation has imaginary roots92

49．Right-hand member not zero94

50．Special case when the right-hand member is not zero96

51．A basic theorem relating to operators97

52．Methods using symbolic operators98

53．Variation of parameters103

54．Simultaneous differential equations106

55．Summary and review exercises108

Chapter 7．Laplace Transforms110

56．Introduction110

57．Definition of a Laplace transform110

58．Some properties of Laplace transforms111

59．Deriving transform relations from given ones114

60．Inverse transforms of products117

61．Transforms of derivatives121

62．Solving differential equations by transforms122

63．Solving systems of differential equations124

64．Resolving a fraction into partial fractions126

65．Fractions having repeated factors in the denominator128

66．Partial fractions．Quadratic factors130

67．Review problems133

Chapter 8．Applications of Linear Equations with Constant Coefficients135

68．Harmonic motion．Damping135

69．Types of damping．Resonance137

70．Forces．Accelerations．Moments139

71．Some fundamental equations of motion140

72．Oscillatory motion141

73．Plane motions of bodies147

74．Kirchhoff＇s current law and electromotive-force law151

75．Simple circuits containing constant electromotive force153

76．Simple circuits containing a sinusoidal electromotive force154

77．Resonance155

78．Applications of Kirchhoff＇s laws to networks156

79．Review problems161

Chapter 9．Miscellaneous Differential Equations of Order Higher than the First163

80．Reduction of order by substitution163

81．Dependent variable absent163

82．Independent variable absent165

83．Method based on factorization of the operator166

84．Euler＇s linear equation169

85．Second-order linear equation170

86．Review exercises171

Chapter 10．Applications173

87．Radius of curvature173

88．Cables．The catenary174

89．Equation of elastic curve．Beams176

90．Columns181

91．Motion of a particle in a plane182

92．Review problems185

Chapter 11．Existence Theorems and Applications187

93．Foreword187

94．Replacement of differential equations by a system of the first order and first degree187

95．Existence theorems188

96．Differential equations of the first order and first degree in the unknowns191

97．Total differential equations194

98．Geometrical interpretation198

99．Fields of force in space198

100．Review exercises200

Chapter 12．Solution by Series202

101．Introduction202

102．Integration in series204

103．Solution involving a more general type of series208

104．Indicial equation has roots differing by an integer210

105．The gamma function213

106．Bessel＇s equation215

107．Bessel＇s functions217

108．Expansion of functions in terms of Bessel＇s functions222

109．Legendre＇s functions224

Chapter 13．Numerical Solutions of Differential Equations229

110．Introduction229

111．Methods of successive approximations229

112．Newton＇s interpolation formula231

113．Interpolation234

114．Formulas for approximate integration236

115．Illustration and discussion of formulas(16)to(21)，§114237

116．Halving the interval h of x240

117．Numerical solution of a system of simultaneous equations242

118．The Runge-Kutta method245

Chapter 14．Partial Differential Equations248

119．Introduction248

120．Solution of a partial differential equation248

121．Equations easily integrable250

122．Equations having the form Pp+Qq=R251

123．Finding particular solutions satisfying given conditions254

124．Separation of variables256

125．Hyperbolic，parabolic，elliptic equations258

Chapter 15．Applications of Partial Differential Equations263

126．Fourier series263

127．Cosine series．Sine series267

128．Application to nuclear fission269

129．Vibrations of a string272

130．Vibrations of a rod275

131．Flow of heat276

132．One-dimensional heat flow279

133．Vibrations of a membrane281

134．Telephone，telegraph，and radio equations283

135．Fluid motion286

Answers291

Index313