Differential Equations with Mathematica, Fifth Edition uses the fundamental concepts of the popular platform to solve (analytically, numerically, and/or graphically) differential equations of interest to students, instructors, and scientists.
Mathematica’s diversity makes it particularly well suited to performing calculations encountered when solving many ordinary and partial differential equations. In some cases, Mathematica’s built-in functions can immediately solve a differential equation by providing an explicit, implicit, or numerical solution. In other cases, Mathematica can be used to perform the calculations encountered when solving a differential equation.
Because one goal of elementary differential equations courses is to introduce students to basic methods and algorithms so that they gain proficiency in them, nearly every topic covered this book introduces basic commands, also including typical examples of their application. A study of differential equations relies on concepts from calculus and linear algebra, so this text also includes discussions of relevant commands useful in those areas. In many cases, seeing a solution graphically is most meaningful, so the book relies heavily on Mathematica’s outstanding graphics capabilities.
1. Chapter 1 Introduction to differential equations - p. 1 2. Keywords - p. 1 3. 1.1 Definitions and concepts - p. 1 4. 1.2 Solutions of differential equations - p. 4 5. 1.3 Initial- and boundary-value problems - p. 12 6. 1.4 Direction fields - p. 18 7. Bibliography - p. 28 8. Chapter 2 First-order ordinary differential equations - p. 29 9. Keywords - p. 29 10. 2.1 Theory of first-order equations: a brief discussion - p. 29 11. 2.2 Separation of variables - p. 34 12. 2.3 Homogeneous equations - p. 42 13. 2.4 Exact equations - p. 46 14. 2.5 Linear equations - p. 50 15. 2.6 Numerical approximations of solutions to first-order equations - p. 60 16. 2.6.1 Built-in methods - p. 60 17. 2.6.2 Other numerical methods - p. 63 18. Application: modeling the spread of a disease - p. 77 19. Bibliography - p. 82 20. Chapter 3 Applications of first-order equations - p. 83 21. Keywords - p. 83 22. 3.1 Orthogonal trajectories - p. 83 23. 3.2 Population growth and decay - p. 90 24. 3.3 Newton's law of cooling - p. 103 25. 3.4 Free-falling bodies - p. 108 26. Bibliography - p. 114 27. Chapter 4 Higher-order linear differential equations - p. 115 28. Keywords - p. 115 29. 4.1 Preliminary definitions and notation - p. 115 30. 4.2 Solving homogeneous equations with constant coefficients - p. 131 31. 4.3 Introduction to solving nonhomogeneous equations - p. 141 32. 4.4 Nonhomogeneous equations with constant coefficients: the method of undetermined coefficients - p. 145 33. 4.4.1 Second-order equations - p. 147 34. 4.4.2 Higher-order equations - p. 158 35. 4.5 Nonhomogeneous equations with constant coefficients: variation of parameters - p. 164 36. 4.6 Cauchy–Euler equations - p. 170 37. 4.7 Series solutions - p. 179 38. 4.7.1 Power series solutions about ordinary points - p. 179 39. 4.7.2 Series solutions about regular singular points - p. 189 40. 4.7.3 Method of Frobenius - p. 190 41. 4.8 Nonlinear equations - p. 205 42. Bibliography - p. 219 43. Chapter 5 Applications of higher-order differential equations - p. 221 44. Keywords - p. 221 45. 5.1 Harmonic motion - p. 221 46. 5.1.1 Simple harmonic motion - p. 221 47. 5.1.2 Damped motion - p. 228 48. 5.1.3 Forced motion - p. 238 49. 5.1.4 Soft springs - p. 250 50. 5.1.5 Hard springs - p. 253 51. 5.1.6 Aging springs - p. 254 52. Application: hearing beats and resonance - p. 255 53. 5.2 The pendulum problem - p. 256 54. 5.3 Other applications - p. 265 55. Chapter 6 Systems of ordinary differential equations - p. 283 56. Keywords - p. 283 57. 6.1 Review of matrix algebra and calculus - p. 283 58. 6.2 Systems of equations: preliminary definitions and theory - p. 297 59. 6.2.1 Preliminary theory - p. 301 60. 6.2.2 Linear systems - p. 308 61. 6.3 Homogeneous linear systems with constant coefficients - p. 315 62. 6.3.1 Distinct real eigenvalues - p. 315 63. 6.3.2 Complex conjugate eigenvalues - p. 320 64. 6.3.3 Solving initial-value problems - p. 325 65. 6.3.4 Repeated eigenvalues - p. 327 66. 6.4 Nonhomogeneous first-order systems: undetermined coefficients, variation of parameters, and the matrix exponential - p. 333 67. 6.4.1 Undetermined coefficients - p. 334 68. 6.4.2 Variation of parameters - p. 337 69. 6.4.3 The matrix exponential - p. 342 70. 6.5 Numerical methods - p. 348 71. 6.6 Nonlinear systems, linearization, and classification of equilibrium points - p. 362 72. 6.6.1 Real distinct eigenvalues - p. 363 73. 6.6.2 Repeated eigenvalues - p. 368 74. 6.6.3 Complex conjugate eigenvalues - p. 371 75. 6.6.4 Nonlinear systems - p. 374 76. Bibliography - p. 384 77. Chapter 7 Applications of systems of ordinary differential equations - p. 385 78. Keywords - p. 385 79. 7.1 Mechanical and electrical problems with first-order linear systems - p. 385 80. 7.2 Diffusion and population problems with first-order linear systems - p. 391 81. 7.3 Applications that lead to nonlinear systems - p. 399 82. 7.3.1 Biological systems: predator–prey interactions, the Lotka–Volterra system, and food chains in the chemostat - p. 400 83. 7.3.2 Physical systems: variable damping - p. 413 84. 7.3.3 Differential geometry: curvature - p. 418 85. Bibliography - p. 421 86. Chapter 8 Laplace transform methods - p. 423 87. Keywords - p. 423 88. 8.1 The Laplace transform - p. 423 89. 8.2 The inverse Laplace transform - p. 432 90. 8.3 Solving initial-value problems with the Laplace transform - p. 439 91. 8.4 Laplace transforms of step and periodic functions - p. 444 92. 8.4.1 Piecewise defined functions: the unit step function - p. 444 93. 8.4.2 Solving initial-value problems with piecewise continuous forcing functions - p. 447 94. 8.4.3 Periodic functions - p. 449 95. 8.4.4 Impulse functions: the delta function - p. 457 96. 8.5 The convolution theorem - p. 462 97. 8.6 Applications of Laplace transforms, Part I - p. 466 98. 8.7 Laplace transform methods for systems - p. 478 99. 8.8 Applications of Laplace transforms, Part II - p. 488 100. Chapter 9 Eigenvalue problems and Fourier series - p. 503 101. Keywords - p. 503 102. 9.1 Boundary-value problems, eigenvalue problems, and Sturm–Liouville problems - p. 503 103. 9.2 Fourier sine series and cosine series - p. 510 104. 9.3 Fourier series - p. 518 105. 9.3.1 Fourier series - p. 518 106. 9.3.2 Even, odd, and periodic extensions - p. 525 107. 9.3.3 Differentiation and integration of Fourier series - p. 530 108. 9.3.4 Parseval's equality - p. 534 109. 9.4 Generalized Fourier series - p. 535 110. Chapter 10 Partial differential equations - p. 545 111. Keywords - p. 545 112. 10.1 Introduction to partial differential equations and separation of variables - p. 545 113. 10.2 The one-dimensional heat equation - p. 548 114. 10.3 The one-dimensional wave equation - p. 557 115. 10.4 Problems in two dimensions: Laplace's equation - p. 565 116. 10.5 Two-dimensional problems in a circular region - p. 570 117. Bibliography - p. 585 118. Index - p. 587
Differential Equations with Mathematica, Fifth Edition uses the fundamental concepts of the popular platform to solve (analytically, numerically, and/or graphically) differential equations of interest to students, instructors, and scientists.
Mathematica’s diversity makes it particularly well suited to performing calculations encountered when solving many ordinary and partial differential equations. In some cases, Mathematica’s built-in functions can immediately solve a differential equation by providing an explicit, implicit, or numerical solution. In other cases, Mathematica can be used to perform the calculations encountered when solving a differential equation.
Because one goal of elementary differential equations courses is to introduce students to basic methods and algorithms so that they gain proficiency in them, nearly every topic covered this book introduces basic commands, also including typical examples of their application. A study of differential equations relies on concepts from calculus and linear algebra, so this text also includes discussions of relevant commands useful in those areas. In many cases, seeing a solution graphically is most meaningful, so the book relies heavily on Mathematica’s outstanding graphics capabilities.
1. Chapter 1 Introduction to differential equations - p. 1 2. Keywords - p. 1 3. 1.1 Definitions and concepts - p. 1 4. 1.2 Solutions of differential equations - p. 4 5. 1.3 Initial- and boundary-value problems - p. 12 6. 1.4 Direction fields - p. 18 7. Bibliography - p. 28 8. Chapter 2 First-order ordinary differential equations - p. 29 9. Keywords - p. 29 10. 2.1 Theory of first-order equations: a brief discussion - p. 29 11. 2.2 Separation of variables - p. 34 12. 2.3 Homogeneous equations - p. 42 13. 2.4 Exact equations - p. 46 14. 2.5 Linear equations - p. 50 15. 2.6 Numerical approximations of solutions to first-order equations - p. 60 16. 2.6.1 Built-in methods - p. 60 17. 2.6.2 Other numerical methods - p. 63 18. Application: modeling the spread of a disease - p. 77 19. Bibliography - p. 82 20. Chapter 3 Applications of first-order equations - p. 83 21. Keywords - p. 83 22. 3.1 Orthogonal trajectories - p. 83 23. 3.2 Population growth and decay - p. 90 24. 3.3 Newton's law of cooling - p. 103 25. 3.4 Free-falling bodies - p. 108 26. Bibliography - p. 114 27. Chapter 4 Higher-order linear differential equations - p. 115 28. Keywords - p. 115 29. 4.1 Preliminary definitions and notation - p. 115 30. 4.2 Solving homogeneous equations with constant coefficients - p. 131 31. 4.3 Introduction to solving nonhomogeneous equations - p. 141 32. 4.4 Nonhomogeneous equations with constant coefficients: the method of undetermined coefficients - p. 145 33. 4.4.1 Second-order equations - p. 147 34. 4.4.2 Higher-order equations - p. 158 35. 4.5 Nonhomogeneous equations with constant coefficients: variation of parameters - p. 164 36. 4.6 Cauchy–Euler equations - p. 170 37. 4.7 Series solutions - p. 179 38. 4.7.1 Power series solutions about ordinary points - p. 179 39. 4.7.2 Series solutions about regular singular points - p. 189 40. 4.7.3 Method of Frobenius - p. 190 41. 4.8 Nonlinear equations - p. 205 42. Bibliography - p. 219 43. Chapter 5 Applications of higher-order differential equations - p. 221 44. Keywords - p. 221 45. 5.1 Harmonic motion - p. 221 46. 5.1.1 Simple harmonic motion - p. 221 47. 5.1.2 Damped motion - p. 228 48. 5.1.3 Forced motion - p. 238 49. 5.1.4 Soft springs - p. 250 50. 5.1.5 Hard springs - p. 253 51. 5.1.6 Aging springs - p. 254 52. Application: hearing beats and resonance - p. 255 53. 5.2 The pendulum problem - p. 256 54. 5.3 Other applications - p. 265 55. Chapter 6 Systems of ordinary differential equations - p. 283 56. Keywords - p. 283 57. 6.1 Review of matrix algebra and calculus - p. 283 58. 6.2 Systems of equations: preliminary definitions and theory - p. 297 59. 6.2.1 Preliminary theory - p. 301 60. 6.2.2 Linear systems - p. 308 61. 6.3 Homogeneous linear systems with constant coefficients - p. 315 62. 6.3.1 Distinct real eigenvalues - p. 315 63. 6.3.2 Complex conjugate eigenvalues - p. 320 64. 6.3.3 Solving initial-value problems - p. 325 65. 6.3.4 Repeated eigenvalues - p. 327 66. 6.4 Nonhomogeneous first-order systems: undetermined coefficients, variation of parameters, and the matrix exponential - p. 333 67. 6.4.1 Undetermined coefficients - p. 334 68. 6.4.2 Variation of parameters - p. 337 69. 6.4.3 The matrix exponential - p. 342 70. 6.5 Numerical methods - p. 348 71. 6.6 Nonlinear systems, linearization, and classification of equilibrium points - p. 362 72. 6.6.1 Real distinct eigenvalues - p. 363 73. 6.6.2 Repeated eigenvalues - p. 368 74. 6.6.3 Complex conjugate eigenvalues - p. 371 75. 6.6.4 Nonlinear systems - p. 374 76. Bibliography - p. 384 77. Chapter 7 Applications of systems of ordinary differential equations - p. 385 78. Keywords - p. 385 79. 7.1 Mechanical and electrical problems with first-order linear systems - p. 385 80. 7.2 Diffusion and population problems with first-order linear systems - p. 391 81. 7.3 Applications that lead to nonlinear systems - p. 399 82. 7.3.1 Biological systems: predator–prey interactions, the Lotka–Volterra system, and food chains in the chemostat - p. 400 83. 7.3.2 Physical systems: variable damping - p. 413 84. 7.3.3 Differential geometry: curvature - p. 418 85. Bibliography - p. 421 86. Chapter 8 Laplace transform methods - p. 423 87. Keywords - p. 423 88. 8.1 The Laplace transform - p. 423 89. 8.2 The inverse Laplace transform - p. 432 90. 8.3 Solving initial-value problems with the Laplace transform - p. 439 91. 8.4 Laplace transforms of step and periodic functions - p. 444 92. 8.4.1 Piecewise defined functions: the unit step function - p. 444 93. 8.4.2 Solving initial-value problems with piecewise continuous forcing functions - p. 447 94. 8.4.3 Periodic functions - p. 449 95. 8.4.4 Impulse functions: the delta function - p. 457 96. 8.5 The convolution theorem - p. 462 97. 8.6 Applications of Laplace transforms, Part I - p. 466 98. 8.7 Laplace transform methods for systems - p. 478 99. 8.8 Applications of Laplace transforms, Part II - p. 488 100. Chapter 9 Eigenvalue problems and Fourier series - p. 503 101. Keywords - p. 503 102. 9.1 Boundary-value problems, eigenvalue problems, and Sturm–Liouville problems - p. 503 103. 9.2 Fourier sine series and cosine series - p. 510 104. 9.3 Fourier series - p. 518 105. 9.3.1 Fourier series - p. 518 106. 9.3.2 Even, odd, and periodic extensions - p. 525 107. 9.3.3 Differentiation and integration of Fourier series - p. 530 108. 9.3.4 Parseval's equality - p. 534 109. 9.4 Generalized Fourier series - p. 535 110. Chapter 10 Partial differential equations - p. 545 111. Keywords - p. 545 112. 10.1 Introduction to partial differential equations and separation of variables - p. 545 113. 10.2 The one-dimensional heat equation - p. 548 114. 10.3 The one-dimensional wave equation - p. 557 115. 10.4 Problems in two dimensions: Laplace's equation - p. 565 116. 10.5 Two-dimensional problems in a circular region - p. 570 117. Bibliography - p. 585 118. Index - p. 587
