Differential Equations
and
Linear Algebra
A Course for
Science and Engineering
Volume I: Chapters 1-7
Volume II: Chapters 8-12
by
Grant B. Gustafson
,©1999–2022 by G. B. Gustafson
All rights reserved
ISBN Volume I: 9798705491124
ISBN Volume II: 9798711123651
ii
,Contents
Table of Contents i
Foreword ii
1 Fundamentals 1
1.1 Exponential Modeling . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Exponential Application Library . . . . . . . . . . . . . . . . . 16
1.3 Differential Equations of First Order . . . . . . . . . . . . . . . 31
1.4 Direction Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.5 Phase Line Diagrams . . . . . . . . . . . . . . . . . . . . . . . . 51
1.6 Computing and Existence . . . . . . . . . . . . . . . . . . . . . 64
2 First Order Differential Equations 74
2.1 Quadrature Method . . . . . . . . . . . . . . . . . . . . . . . . 74
2.2 Separable Equations . . . . . . . . . . . . . . . . . . . . . . . . 82
2.3 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 93
2.4 Undetermined Coefficients . . . . . . . . . . . . . . . . . . . . . 104
2.5 Linear Applications . . . . . . . . . . . . . . . . . . . . . . . . . 111
2.6 Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
2.7 Logistic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 142
2.8 Science and Engineering Applications . . . . . . . . . . . . . . 147
2.9 Exact Equations and Level Curves . . . . . . . . . . . . . . . . 162
2.10 Special equations . . . . . . . . . . . . . . . . . . . . . . . . . . 167
3 Linear Algebraic Equations No Matrices 174
3.1 Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . 174
3.2 Filmstrips and Toolkit Sequences . . . . . . . . . . . . . . . . . 185
3.3 General Solution Theory . . . . . . . . . . . . . . . . . . . . . . 194
i
, CONTENTS
3.4 Basis, Dimension, Nullity and Rank . . . . . . . . . . . . . . . 207
3.5 Answer Check, Proofs and Details . . . . . . . . . . . . . . . . 218
4 Numerical Methods with Applications 225
4.1 Solving y′ = F (x) Numerically . . . . . . . . . . . . . . . . . . 225
4.2 Solving y′ = f (x, y) Numerically . . . . . . . . . . . . . . . . . 238
4.3 Error in Numerical Methods . . . . . . . . . . . . . . . . . . . . 247
4.4 Computing π, ln 2 and e . . . . . . . . . . . . . . . . . . . . . . 254
4.5 Earth to the Moon . . . . . . . . . . . . . . . . . . . . . . . . . 260
4.6 Skydiving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
4.7 Lunar Lander . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
4.8 Comets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
4.9 Fish Farming . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
5 Linear Algebra 293
5.1 Vectors and Matrices . . . . . . . . . . . . . . . . . . . . . . . . 294
5.2 Matrix Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 321
5.3 Determinants and Cramer’s Rule . . . . . . . . . . . . . . . . . 343
5.4 Vector Spaces, Independence, Basis . . . . . . . . . . . . . . . . 369
5.5 Basis, Dimension and Rank . . . . . . . . . . . . . . . . . . . . 405
6 Scalar Linear Differential Equations 430
6.1 Linear 2nd Order Constant . . . . . . . . . . . . . . . . . . . . 430
6.2 Continuous Coefficient Theory . . . . . . . . . . . . . . . . . . 442
6.3 Higher Order Linear Constant Equations . . . . . . . . . . . . . 451
6.4 Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . 464
6.5 Undetermined Coefficients . . . . . . . . . . . . . . . . . . . . . 470
6.6 Undamped Mechanical Vibrations . . . . . . . . . . . . . . . . 490
6.7 Forced and Damped Vibrations . . . . . . . . . . . . . . . . . . 506
6.8 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528
6.9 Kepler’s laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
7 Topics in Linear Differential Equations 551
7.1 Higher Order Homogeneous . . . . . . . . . . . . . . . . . . . . 551
7.2 Differential Operators . . . . . . . . . . . . . . . . . . . . . . . 557
7.3 Higher Order Non-Homogeneous . . . . . . . . . . . . . . . . . 560
7.4 Cauchy-Euler Equation . . . . . . . . . . . . . . . . . . . . . . 566
ii
and
Linear Algebra
A Course for
Science and Engineering
Volume I: Chapters 1-7
Volume II: Chapters 8-12
by
Grant B. Gustafson
,©1999–2022 by G. B. Gustafson
All rights reserved
ISBN Volume I: 9798705491124
ISBN Volume II: 9798711123651
ii
,Contents
Table of Contents i
Foreword ii
1 Fundamentals 1
1.1 Exponential Modeling . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Exponential Application Library . . . . . . . . . . . . . . . . . 16
1.3 Differential Equations of First Order . . . . . . . . . . . . . . . 31
1.4 Direction Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.5 Phase Line Diagrams . . . . . . . . . . . . . . . . . . . . . . . . 51
1.6 Computing and Existence . . . . . . . . . . . . . . . . . . . . . 64
2 First Order Differential Equations 74
2.1 Quadrature Method . . . . . . . . . . . . . . . . . . . . . . . . 74
2.2 Separable Equations . . . . . . . . . . . . . . . . . . . . . . . . 82
2.3 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 93
2.4 Undetermined Coefficients . . . . . . . . . . . . . . . . . . . . . 104
2.5 Linear Applications . . . . . . . . . . . . . . . . . . . . . . . . . 111
2.6 Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
2.7 Logistic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 142
2.8 Science and Engineering Applications . . . . . . . . . . . . . . 147
2.9 Exact Equations and Level Curves . . . . . . . . . . . . . . . . 162
2.10 Special equations . . . . . . . . . . . . . . . . . . . . . . . . . . 167
3 Linear Algebraic Equations No Matrices 174
3.1 Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . 174
3.2 Filmstrips and Toolkit Sequences . . . . . . . . . . . . . . . . . 185
3.3 General Solution Theory . . . . . . . . . . . . . . . . . . . . . . 194
i
, CONTENTS
3.4 Basis, Dimension, Nullity and Rank . . . . . . . . . . . . . . . 207
3.5 Answer Check, Proofs and Details . . . . . . . . . . . . . . . . 218
4 Numerical Methods with Applications 225
4.1 Solving y′ = F (x) Numerically . . . . . . . . . . . . . . . . . . 225
4.2 Solving y′ = f (x, y) Numerically . . . . . . . . . . . . . . . . . 238
4.3 Error in Numerical Methods . . . . . . . . . . . . . . . . . . . . 247
4.4 Computing π, ln 2 and e . . . . . . . . . . . . . . . . . . . . . . 254
4.5 Earth to the Moon . . . . . . . . . . . . . . . . . . . . . . . . . 260
4.6 Skydiving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
4.7 Lunar Lander . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
4.8 Comets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
4.9 Fish Farming . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
5 Linear Algebra 293
5.1 Vectors and Matrices . . . . . . . . . . . . . . . . . . . . . . . . 294
5.2 Matrix Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 321
5.3 Determinants and Cramer’s Rule . . . . . . . . . . . . . . . . . 343
5.4 Vector Spaces, Independence, Basis . . . . . . . . . . . . . . . . 369
5.5 Basis, Dimension and Rank . . . . . . . . . . . . . . . . . . . . 405
6 Scalar Linear Differential Equations 430
6.1 Linear 2nd Order Constant . . . . . . . . . . . . . . . . . . . . 430
6.2 Continuous Coefficient Theory . . . . . . . . . . . . . . . . . . 442
6.3 Higher Order Linear Constant Equations . . . . . . . . . . . . . 451
6.4 Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . 464
6.5 Undetermined Coefficients . . . . . . . . . . . . . . . . . . . . . 470
6.6 Undamped Mechanical Vibrations . . . . . . . . . . . . . . . . 490
6.7 Forced and Damped Vibrations . . . . . . . . . . . . . . . . . . 506
6.8 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528
6.9 Kepler’s laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
7 Topics in Linear Differential Equations 551
7.1 Higher Order Homogeneous . . . . . . . . . . . . . . . . . . . . 551
7.2 Differential Operators . . . . . . . . . . . . . . . . . . . . . . . 557
7.3 Higher Order Non-Homogeneous . . . . . . . . . . . . . . . . . 560
7.4 Cauchy-Euler Equation . . . . . . . . . . . . . . . . . . . . . . 566
ii
