Elementary differential equations and boundary value problems / William E. Boyce and Richard C. Di Prima.
Editor: New York : Wiley, c1969Edición: 2nd edDescripción: xiv, 533, A-43, I-8 p. : il. ; 24 cmISBN: 0471093319Otra clasificación: 34-011. INTRODUCTION [1] 1.1 Ordinary Differential Equations [2] 1.2 Historical Remarks [5] 2. FIRST ORDER DIFFERENTIAL EQUATIONS [10] 2.1 Linear Equations [10] 2.2 Further Discussion of Linear Equations [17] 2.3 Nonlinear Equations [21] 2.4 Separable Equations [29] 2.5 Exact Equations [34] 2.6 Integrating Factors [39] 2.7 Homogeneous Equations [43] 2.8 Miscellaneous Problems [47] 2.9 Applications of First Order Equations [51] 2.10 Elementary Mechanics [61] 2.11 The Existence and Uniqueness Theorem [69] 2.12 The Existence Theorem from a More Modern Viewpoint [80] Appendix. Derivation of Equation of Motion of Body with Variable Mass [82] 3. SECOND ORDER LINEAR EQUATIONS [85] 3.1 Introduction [85] 3.2 Fundamental Solutions of the Homogeneous Equation [90] 3.3 Linear Independence [99] 3.4 Reduction of Order [103] 3.5 Homogeneous Equations with Constant Coefficients [106] 3.5.1 Complex Roots [110] 3.6 The Nonhomogeneous Problem [114] 3.6.1 The Method of Undetermined Coefficients [117] 3.6.2 The Method of Variation of Parameters [124] 3.7 Mechanical Vibrations [129] 3.7.1 Free Vibrations [133] 3.7.2 Forced Vibrations [138] 3.8 Electrical Networks [142] 4. SERIES SOLUTIONS OF SECOND ORDER LINEAR EQUATIONS [148] 4.1 Introduction. Review of Power Series [148] 4.2 Series Solutions near an Ordinary Point, Part I [152] 4.2.1 Series Solutions near an Ordinary Point, Part II [159] 4.3 Regular Singular Points [167] 4.4 Euler Equations [171] 4.5 Series Solutions near a Regular Singular Point, Part I [177] 4.5.1 Series Solutions near a Regular Singular Point, Part II [183] *4.6 Series Solutions near a Regular Singular Point; r1 = r2 and r1 — r2 = N [188] *4.7 Bessel’s Equation [191] 5. HIGHER ORDER LINEAR EQUATIONS [202] 5.1 Introduction [202] 5.2 General Theory of //th Order Linear Equations [204] 5.3 The Homogeneous Equation with Constant Coefficients [208] 5.4 The Method of Undetermined Coefficients [215] 5.5 The Method of Variation of Parameters [218] 6. THE LAPLACE TRANSFORM [222] 6.1 Introduction. Definition of the Laplace Transform [222] 6.2 Solution of Initial Value Problems [228] 6.3 Step Functions [237] 6.3.1 A Differential Equation with a Discontinuous Forcing Function [245] 6.4 Impulse Functions [249] 6.5 The Convolution Integral [253] 6.6 General Discussion and Summary [258] 7. SYSTEMS OF FIRST ORDER EQUATIONS [261] 7.1 Introduction [261] 7.2 Solution of Linear Systems by Elimination [266] 7.3 Review of Matrices [272] 7.4 Basic Theory of Systems of Linear First Order Equations [282] 7.5 Linear Homogeneous Systems with Constant Coefficients [288] 7.6 Inverses, Eigenvalues, and Eigenvectors [294] 7.7 Fundamental Matrices [300] 7.8 Complex Roots [307] 7.9 Repeated Roots [312] 7.10 Nonhomogeneous Linear Systems [317] 7.11 The Laplace Transform for Systems of Equations [322] 8. NUMERICAL METHODS [328] 8.1 Introduction [328] 8.2 The Euler or Tangent Line Method [330] 8.3 The Error [338] 8.4 An Improved Euler Method [345] 8.5 The Three-Term Taylor Series Method [350] 8.6 The Runge-Kutta Method [353] 8.7 Some Difficulties with Numerical Methods [357] 8.8 A Multistep Method [361] 8.9 Systems of First Order Equations [365] 9. NONLINEAR DIFFERENTIAL EQUATIONS AND STABILITY [369] 9.1 Introduction [369] 9.2 Solutions and Trajectories [375] 9.3 The Phase Plane: The Linear System [382] 9.4 Stability; Almost Linear Systems [392] 9.5 Liapounov's Second Method [403] 9.6 Periodic Solutions and Limit Cycles [413] 10. PARTIAL DIFFERENTIAL EQUATIONS AND FOURIER SERIES [422] 10.1 Introduction [422] 10.2 Heat Conduction and Separation of Variables [423] 10.3 Fourier Series [431] 10.4 The Fourier Theorem [438] 10.5 Even and Odd Functions [445] 10.6 Solution of Other Heat Conduction Problems [453] 10.7 The Wave Equation; Vibrations of an Elastic String [460] 10.8 Laplace’s Equation [470] Appendix A. Derivation of the Heat Conduction Equation [478] Appendix B. Derivation of the Wave Equation [481] 11. BOUNDARY VALUE PROBLEMS AND STURM-LIOUVILLE THEORY [485] 11.1 Introduction [485] 11.2 Linear Homogeneous Boundary Value Problems: Eigenvalues and Eigenfunctions [486] 11.3 Sturm-Liouville Boundary Value Problems [493] 11.4 Solution of Nonhomogeneous Boundary Value Problems [501] 11.5 Singular Sturm-Liouville Problems [511] 11.6 Further Remarks on the Method of Separation of Variables; A Bessel Series Expansion [518] 11.7 Series of Orthogonal Functions; Mean Convergence [525] 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 [10] --
2.1 Linear Equations [10] --
2.2 Further Discussion of Linear Equations [17] --
2.3 Nonlinear Equations [21] --
2.4 Separable Equations [29] --
2.5 Exact Equations [34] --
2.6 Integrating Factors [39] --
2.7 Homogeneous Equations [43] --
2.8 Miscellaneous Problems [47] --
2.9 Applications of First Order Equations [51] --
2.10 Elementary Mechanics [61] --
2.11 The Existence and Uniqueness Theorem [69] --
2.12 The Existence Theorem from a More Modern Viewpoint [80] --
Appendix. Derivation of Equation of Motion of Body with Variable Mass [82] --
3. SECOND ORDER LINEAR EQUATIONS [85] --
3.1 Introduction [85] --
3.2 Fundamental Solutions of the Homogeneous Equation [90] --
3.3 Linear Independence [99] --
3.4 Reduction of Order [103] --
3.5 Homogeneous Equations with Constant Coefficients [106] --
3.5.1 Complex Roots [110] --
3.6 The Nonhomogeneous Problem [114] --
3.6.1 The Method of Undetermined Coefficients [117] --
3.6.2 The Method of Variation of Parameters [124] --
3.7 Mechanical Vibrations [129] --
3.7.1 Free Vibrations [133] --
3.7.2 Forced Vibrations [138] --
3.8 Electrical Networks [142] --
4. SERIES SOLUTIONS OF SECOND ORDER LINEAR EQUATIONS [148] --
4.1 Introduction. Review of Power Series [148] --
4.2 Series Solutions near an Ordinary Point, Part I [152] --
4.2.1 Series Solutions near an Ordinary Point, Part II [159] --
4.3 Regular Singular Points [167] --
4.4 Euler Equations [171] --
4.5 Series Solutions near a Regular Singular Point, Part I [177] --
4.5.1 Series Solutions near a Regular Singular Point, Part II [183] --
*4.6 Series Solutions near a Regular Singular Point; r1 = r2 and r1 — r2 = N [188] --
*4.7 Bessel’s Equation [191] --
5. HIGHER ORDER LINEAR EQUATIONS [202] --
5.1 Introduction [202] --
5.2 General Theory of //th Order Linear Equations [204] --
5.3 The Homogeneous Equation with Constant Coefficients [208] --
5.4 The Method of Undetermined Coefficients [215] --
5.5 The Method of Variation of Parameters [218] --
6. THE LAPLACE TRANSFORM [222] --
6.1 Introduction. Definition of the Laplace Transform [222] --
6.2 Solution of Initial Value Problems [228] --
6.3 Step Functions [237] --
6.3.1 A Differential Equation with a Discontinuous Forcing Function [245] --
6.4 Impulse Functions [249] --
6.5 The Convolution Integral [253] --
6.6 General Discussion and Summary [258] --
7. SYSTEMS OF FIRST ORDER EQUATIONS [261] --
7.1 Introduction [261] --
7.2 Solution of Linear Systems by Elimination [266] --
7.3 Review of Matrices [272] --
7.4 Basic Theory of Systems of Linear First Order Equations [282] --
7.5 Linear Homogeneous Systems with Constant Coefficients [288] --
7.6 Inverses, Eigenvalues, and Eigenvectors [294] --
7.7 Fundamental Matrices [300] --
7.8 Complex Roots [307] --
7.9 Repeated Roots [312] --
7.10 Nonhomogeneous Linear Systems [317] --
7.11 The Laplace Transform for Systems of Equations [322] --
8. NUMERICAL METHODS [328] --
8.1 Introduction [328] --
8.2 The Euler or Tangent Line Method [330] --
8.3 The Error [338] --
8.4 An Improved Euler Method [345] --
8.5 The Three-Term Taylor Series Method [350] --
8.6 The Runge-Kutta Method [353] --
8.7 Some Difficulties with Numerical Methods [357] --
8.8 A Multistep Method [361] --
8.9 Systems of First Order Equations [365] --
9. NONLINEAR DIFFERENTIAL EQUATIONS AND STABILITY [369] --
9.1 Introduction [369] --
9.2 Solutions and Trajectories [375] --
9.3 The Phase Plane: The Linear System [382] --
9.4 Stability; Almost Linear Systems [392] --
9.5 Liapounov's Second Method [403] --
9.6 Periodic Solutions and Limit Cycles [413] --
10. PARTIAL DIFFERENTIAL EQUATIONS AND FOURIER SERIES [422] --
10.1 Introduction [422] --
10.2 Heat Conduction and Separation of Variables [423] --
10.3 Fourier Series [431] --
10.4 The Fourier Theorem [438] --
10.5 Even and Odd Functions [445] --
10.6 Solution of Other Heat Conduction Problems [453] --
10.7 The Wave Equation; Vibrations of an Elastic String [460] --
10.8 Laplace’s Equation [470] --
Appendix A. Derivation of the Heat Conduction Equation [478] --
Appendix B. Derivation of the Wave Equation [481] --
11. BOUNDARY VALUE PROBLEMS AND --
STURM-LIOUVILLE THEORY [485] --
11.1 Introduction [485] --
11.2 Linear Homogeneous Boundary Value Problems: Eigenvalues and Eigenfunctions [486] --
11.3 Sturm-Liouville Boundary Value Problems [493] --
11.4 Solution of Nonhomogeneous Boundary Value Problems [501] --
11.5 Singular Sturm-Liouville Problems [511] --
11.6 Further Remarks on the Method of Separation of Variables; A Bessel Series Expansion [518] --
11.7 Series of Orthogonal Functions; Mean Convergence [525] --
ANSWERS TO PROBLEMS A-1 --
INDEX 1-1 --
MR, REVIEW #
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