## Differential Equations: An Introduction with Mathematica®Goals and Emphasis of the Book Mathematicians have begun to find productive ways to incorporate computing power into the mathematics curriculum. There is no attempt here to use computing to avoid doing differential equations and linear algebra. The goal is to make some first ex plorations in the subject accessible to students who have had one year of calculus. Some of the sciences are now using the symbol-manipulative power of Mathemat ica to make more of their subject accessible. This book is one way of doing so for differential equations and linear algebra. I believe that if a student's first exposure to a subject is pleasant and exciting, then that student will seek out ways to continue the study of the subject. The theory of differential equations and of linear algebra permeates the discussion. Every topic is supported by a statement of the theory. But the primary thrust here is obtaining solutions and information about solutions, rather than proving theorems. There are other courses where proving theorems is central. The goals of this text are to establish a solid understanding of the notion of solution, and an appreciation for the confidence that the theory gives during a search for solutions. Later the student can have the same confidence while personally developing the theory. |

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### Contents

About Differential Equations | 1 |

11 Numerical Methods | 8 |

12 Uniqueness Considerations | 17 |

13 Differential Inclusions Optional | 22 |

Linear Algebra | 26 |

21 Familiar Linear Spaces | 31 |

23 Differential Equations from Solutions | 44 |

24 Characteristic Value Problems | 48 |

The Laplace Transform | 210 |

71 The Laplace Transform | 211 |

72 Properties of the Laplace Transform | 214 |

73 The Inverse Laplace Transform | 225 |

74 Discontinous Functions and Their Transforms | 230 |

HigherOrder Differential Equations with Variable Coefficients | 240 |

81 CauchyEuler Differential Equations | 241 |

82 Obtaining a Second Solution | 251 |

FirstOrder Differential Equations | 52 |

32 Linear Equations by Mathematica | 57 |

33 Exact Equations | 59 |

34 Variables Separable | 69 |

35 Homogeneous Nonlinear Differential Equations | 75 |

36 Bernoulli and Riccati Differential Equations Optional | 79 |

37 Clairaut Differential Equations Optional | 86 |

Applications of FirstOrder Equations | 90 |

42 Linear Applications | 95 |

43 Nonlinear Applications | 117 |

HigherOrder Linear Differential Equations | 129 |

51 The Fundamental Theorem | 130 |

52 Homogeneous SecondOrder Linear Constant Coefficients | 139 |

53 HigherOrder Constant Coefficients Homogeneous | 152 |

54 The Method of Undetermined Coefficients | 160 |

55 Variation of Parameters | 171 |

Applications of SecondOrder Equations | 179 |

62 Damped Harmonic Motion | 190 |

63 Forced Oscillation | 197 |

64 Simple Electronic Circuits | 202 |

65 Two Nonlinear Examples Optional | 206 |

83 Sums Products and Recursion Relations | 253 |

84 Series Solutions of Differential Equations | 262 |

85 Series Solutions About Ordinary Points | 269 |

86 Series Solution About Regular Singular Points | 277 |

87 Important Classical Differential Equations and Functions | 293 |

Differential Systems Theory | 297 |

91 Reduction to FirstOrder Systems | 302 |

92 Theory of FirstOrder Systems | 309 |

93 FirstOrder Constant Coefficients Systems | 317 |

94 Repeated and Complex Roots | 331 |

95 Nonhomogeneous Equations and BoundaryValue Problems | 344 |

96 CauchyEuler Systems | 358 |

Differential Systems Applications | 369 |

102 Phase Portraits | 384 |

103 Two Nonlinear Examples Optional | 404 |

104 Defective Systems of FirstOrder Differential Equations Optional | 410 |

105 Solution of Linear Systems by Laplace Transforms Optional | 416 |

423 | |

426 | |

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annihilator apply basis behavior calculation Cauchy-Euler equations chapter characteristic polynomial characteristic roots characteristic vectors Check coefficient matrix complete solution constant coefficients constant solutions container corresponding curves decay defined definition derivatives determine diagonalizable differential inclusion differential operator differential system DSolve dx3 dx2 dx dy/dt dy/dx equa Euler's method Example Exercises exponential expression function fundamental matrix fundamental solution given higher-order homogeneous indicial equation initial conditions initial value problem integrating factor interval inverse kernel Laplace transform linear algebra linear differential equation linear space linearly independent solutions manually Mathematica Mathematics matrix exponential method motion multiplicity nonhomogeneous nonzero notebook obtain particular solution plot population power series recursion relation regular singular point salt second-order series solution Show Simplify solution that passes solve substitution Suppose Theorem theory tion undetermined coefficients unique solution variables velocity verify Wronskian

### Popular passages

Page 423 - REFERENCES 1. M. Abramowitz and I. Stegun, Handbook of mathematical functions, Dover, New York, 1965.