Elementary differential equations and boundary value problems / William E. Boyce and Richard C. DiPrima.
Editor: New York : Wiley, c1977Edición: 3rd edDescripción: 652 p. en varias paginaciones : il. ; 24 cmISBN: 0471093343Tema(s): Differential equations | Boundary value problemsOtra clasificación: 34-011. INTRODUCTION [1] 1.1 Ordinary Differential Equations [2] 1.2 Historical Remarks [5] 2. FIRST ORDER DIFFERENTIAL EQUATIONS [11] 2.1 Linear Equations [11] 2.2 Further Discussion of Linear Equations [19] 2.3 Nonlinear Equations [24] 2.4 Separable Equations [31] 2.5 Exact Equations [36] 2.6 Integrating Factors [42] 2.7 Homogeneous Equations [45] 2.8 Miscellaneous Problems [49] 2.9 Applications of First Order Equations [51] 2.10 Elementary Mechanics [62] *2.11 The Existence and Uniqueness Theorem [70] Appendix. Derivation of Equation of Motion of a Body with Variable Mass [80] 3. SECOND ORDER LINEAR EQUATIONS [82] 3.1 Introduction [82] 3.2 Fundamental Solutions of the Homogeneous Equation [87] 3.3 Linear Independence [96] 3.4 Reduction of Order [99] 3.5 Homogeneous Equations with Constant Coefficients [103] 3.5.1 Complex Roots [107] 3.6 The Nonhomogeneous Problem [112] 3.6.1 The Method of Undetermined Coefficients [115] 3.6.2 The Method of Variation of Parameters [121] 3.7 Mechanical Vibrations [127] 3.7.1 Free Vibrations [130] 3.7.2 Forced Vibrations [135] 3.8 Electrical Networks [139] 4. SERIES SOLUTIONS OF SECOND ORDER LINEAR EQUATIONS [145] 4.1 Introduction. Review of Power Series [145] 4.2 Series Solutions Near an Ordinary Point, Part I [152] 4.2.1 Series Solutions Near an Ordinary Point, Part II [161] 4.3 Regular Singular Points [169] 4.4 Euler Equations [174] 4.5 Series Solutions Near a Regular Singular Point, Part I [179] 4.5.1 Series Solutions Near a Regular Singular Point, Part II [186] *4.6 Series Solutions Near a Regular Singular Point; r1= r2 and r1 — r2 = N [192] *4.7 Bessel’s Equation [195] 5. HIGHER ORDER LINEAR EQUATIONS [207] 5.1 Introduction [207] 5.2 General Theory of nth Order Linear Equations [208] 5.3 Homogeneous Equations with Constant Coefficients [213] 5.4 The Method of Undetermined Coefficients [219] 5.5 The Method of Variation of Parameters [222] 6. THE LAPLACE TRANSFORM [226] 6.1 Introduction. Definition of the Laplace Transform [226] 6.2 Solution of Initial Value Problems [233] 6.3 Step Functions [242] 6.3.1 A Differential Equation with a Discontinuous Forcing Function [249] 6.4 Impulse Functions [252] 6.5 The Convolution Integral [257] 6.6 General Discussion and Summary [263] 7. SYSTEMS OF FIRST ORDER LINEAR EQUATIONS [265] 7.1 Introduction [265] 7.2 Solution of Linear Systems by Elimination [272] 7.3 Review of Matrices [277] 7.4 Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors [287] 7.5 Basic Theory of Systems of First Order Linear Equations [299] 7.6 Homogeneous Linear Systems with Constant Coefficients [304] 7.7 Complex Eigenvalues [312] 7.8 Repeated Eigenvalues [317] 7.9 Fundamental Matrices [324] 7.10 Nonhomogeneous Linear Systems [329] NUMERICAL METHODS [336] 8.1 Introduction [336] 8.2 The Euler or Tangent Line Method [338] 8.3 The Error [344] 8.4 An Improved Euler Method [351] 8.5 The Three-Term Taylor Series Method [356] 8.6 The Runge-Kutta Method [358] 8.7 Some Difficulties with Numerical Methods [363] 8.8 A Multistep Method [367] 8.9 Systems of First Order Equations [374] 9. NONLINEAR DIFFERENTIAL EQUATIONS AND STABILITY [378] 9.1 Introduction [386] 9.2 Solutions of Autonomous Systems [378] 9.3 The Phase Plane; Linear Systems [397] 9.4 Stability; Almost Linear Systems [409] 9.5 Competing Species and Predator-Prey Problems [421] 9.6 Liapounov’s Second Method [434] *9.7 Periodic Solutions and Limit Cycles [443] 10. PARTIAL DIFFERENTIAL EQUATIONS AND FOURIER SERIES [452] 10.1 Introduction [452] 10.2 Heat Conduction and Separation of Variables [453] 10.3 Fourier Series [461] 10.4 The Fourier Theorem [470] 10.5 Even and Odd Functions [476] 10.6 Solution of Other Heat Conduction Problems [483] 10.7 The Wave Equation; Vibrations of an Elastic String [491] 10.8 Laplace’s Equation [503] Appendix A. Derivation of the Heat Conduction Equation [511] Appendix B. Derivation of the Wave Equation [515] 11. BOUNDARY VALUE PROBLEMS AND STURM-LIOUVILLE THEORY [519] 11.1 Introduction [519] 11.2 Linear Homogeneous Boundary Value Problems; Eigenvalues and Eigenfunctions [523] 11.3 Sturm-Liouville Boundary Value Problems [531] 11.4 Nonhomogeneous Boundary Value Problems [544] *11.5 Singular Sturm-Liouville Problems [560] *11.6 Further Remarks on the Method of Separation of Variables; A Bessel Series Expansion [568] ♦11.7 Series of Orthogonal Functions; Mean Convergence [574] ANSWERS TO PROBLEMS A-1 INDEX 1-1
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1. INTRODUCTION [1] --
1.1 Ordinary Differential Equations [2] --
1.2 Historical Remarks [5] --
2. FIRST ORDER DIFFERENTIAL EQUATIONS [11] --
2.1 Linear Equations [11] --
2.2 Further Discussion of Linear Equations [19] --
2.3 Nonlinear Equations [24] --
2.4 Separable Equations [31] --
2.5 Exact Equations [36] --
2.6 Integrating Factors [42] --
2.7 Homogeneous Equations [45] --
2.8 Miscellaneous Problems [49] --
2.9 Applications of First Order Equations [51] --
2.10 Elementary Mechanics [62] --
*2.11 The Existence and Uniqueness Theorem [70] --
Appendix. Derivation of Equation of Motion of a Body with Variable Mass [80] --
3. SECOND ORDER LINEAR EQUATIONS [82] --
3.1 Introduction [82] --
3.2 Fundamental Solutions of the Homogeneous Equation [87] --
3.3 Linear Independence [96] --
3.4 Reduction of Order [99] --
3.5 Homogeneous Equations with Constant Coefficients [103] --
3.5.1 Complex Roots [107] --
3.6 The Nonhomogeneous Problem [112] --
3.6.1 The Method of Undetermined Coefficients [115] --
3.6.2 The Method of Variation of Parameters [121] --
3.7 Mechanical Vibrations [127] --
3.7.1 Free Vibrations [130] --
3.7.2 Forced Vibrations [135] --
3.8 Electrical Networks [139] --
4. SERIES SOLUTIONS OF SECOND ORDER LINEAR EQUATIONS [145] --
4.1 Introduction. Review of Power Series [145] --
4.2 Series Solutions Near an Ordinary Point, Part I [152] --
4.2.1 Series Solutions Near an Ordinary Point, Part II [161] --
4.3 Regular Singular Points [169] --
4.4 Euler Equations [174] --
4.5 Series Solutions Near a Regular Singular Point, Part I [179] --
4.5.1 Series Solutions Near a Regular Singular Point, Part II [186] --
*4.6 Series Solutions Near a Regular Singular Point; r1= r2 and r1 — r2 = N [192] --
*4.7 Bessel’s Equation [195] --
5. HIGHER ORDER LINEAR EQUATIONS [207] --
5.1 Introduction [207] --
5.2 General Theory of nth Order Linear Equations [208] --
5.3 Homogeneous Equations with Constant Coefficients [213] --
5.4 The Method of Undetermined Coefficients [219] --
5.5 The Method of Variation of Parameters [222] --
6. THE LAPLACE TRANSFORM [226] --
6.1 Introduction. Definition of the Laplace Transform [226] --
6.2 Solution of Initial Value Problems [233] --
6.3 Step Functions [242] --
6.3.1 A Differential Equation with a Discontinuous Forcing Function [249] --
6.4 Impulse Functions [252] --
6.5 The Convolution Integral [257] --
6.6 General Discussion and Summary [263] --
7. SYSTEMS OF FIRST ORDER LINEAR EQUATIONS [265] --
7.1 Introduction [265] --
7.2 Solution of Linear Systems by Elimination [272] --
7.3 Review of Matrices [277] --
7.4 Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors [287] --
7.5 Basic Theory of Systems of First Order Linear Equations [299] --
7.6 Homogeneous Linear Systems with Constant Coefficients [304] --
7.7 Complex Eigenvalues [312] --
7.8 Repeated Eigenvalues [317] --
7.9 Fundamental Matrices [324] --
7.10 Nonhomogeneous Linear Systems [329] --
NUMERICAL METHODS [336] --
8.1 Introduction [336] --
8.2 The Euler or Tangent Line Method [338] --
8.3 The Error [344] --
8.4 An Improved Euler Method [351] --
8.5 The Three-Term Taylor Series Method [356] --
8.6 The Runge-Kutta Method [358] --
8.7 Some Difficulties with Numerical Methods [363] --
8.8 A Multistep Method [367] --
8.9 Systems of First Order Equations [374] --
9. NONLINEAR DIFFERENTIAL EQUATIONS AND STABILITY [378] --
9.1 Introduction [386] --
9.2 Solutions of Autonomous Systems [378] --
9.3 The Phase Plane; Linear Systems [397] --
9.4 Stability; Almost Linear Systems [409] --
9.5 Competing Species and Predator-Prey Problems [421] --
9.6 Liapounov’s Second Method [434] --
*9.7 Periodic Solutions and Limit Cycles [443] --
10. PARTIAL DIFFERENTIAL EQUATIONS AND FOURIER SERIES [452] --
10.1 Introduction [452] --
10.2 Heat Conduction and Separation of Variables [453] --
10.3 Fourier Series [461] --
10.4 The Fourier Theorem [470] --
10.5 Even and Odd Functions [476] --
10.6 Solution of Other Heat Conduction Problems [483] --
10.7 The Wave Equation; Vibrations of an Elastic String [491] --
10.8 Laplace’s Equation [503] --
Appendix A. Derivation of the Heat Conduction Equation [511] --
Appendix B. Derivation of the Wave Equation [515] --
11. BOUNDARY VALUE PROBLEMS AND STURM-LIOUVILLE THEORY [519] --
11.1 Introduction [519] --
11.2 Linear Homogeneous Boundary Value Problems; Eigenvalues and Eigenfunctions [523] --
11.3 Sturm-Liouville Boundary Value Problems [531] --
11.4 Nonhomogeneous Boundary Value Problems [544] --
*11.5 Singular Sturm-Liouville Problems [560] --
*11.6 Further Remarks on the Method of Separation of Variables; A Bessel Series Expansion [568] --
♦11.7 Series of Orthogonal Functions; Mean Convergence [574] --
ANSWERS TO PROBLEMS A-1 --
INDEX 1-1 --
MR, REVIEW #
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