Solution Manual For Differential Equations and Linear Algebra : A Course for Science and Engineering Volume 1 & 2 by Prof Grant B. Gustafson All Chapters 1- 7 & 8-12

Solution Manual For Differential Equations and Linear Algebra : A Course for Science and Engineering Volume 1 & 2 by Prof Grant B. Gustafson All Chapters 1- 7 & 8-12 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 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 CONTENTS 7.5 Variation of Parameters Revisited . . . . . . . . . . . . . . . . . 569 7.6 Undetermined Coefficients Library . . . . . . . . . . . . . . . . 574 8 Laplace Transform 591 8.1 Laplace Method Introduction . . . . . . . . . . . . . . . . . . . 592 8.2 Laplace Integral Table . . . . . . . . . . . . . . . . . . . . . . . 601 8.3 Laplace Transform Rules . . . . . . . . . . . . . . . . . . . . . . 609 8.4 Heaviside’s Partial Fraction Method . . . . . . . . . . . . . . . 620 8.5 Transform Properties . . . . . . . . . . . . . . . . . . . . . . . . 637 8.6 Heaviside Step and Dirac Impulse . . . . . . . . . . . . . . . . . 644 8.7 Laplace Table Derivations . . . . . . . . . . . . . . . . . . . . . 650 8.8 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654 9 Eigenanalysis 667 9.1 Matrix Eigenanalysis . . . . . . . . . . . . . . . . . . . . . . . . 667 9.2 Eigenanalysis Applications . . . . . . . . . . . . . . . . . . . . . 703 9.3 Advanced Topics in Linear Algebra . . . . . . . . . . . . . . . . 720 10 Phase Plane Methods 750 10.1 Planar Autonomous Systems . . . . . . . . . . . . . . . . . . . 751 10.2 Planar Constant Linear Systems . . . . . . . . . . . . . . . . . 767 10.3 Planar Almost Linear Systems . . . . . . . . . . . . . . . . . . 780 10.4 Biological Models . . . . . . . . . . . . . . . . . . . . . . . . . . 790 10.5 Mechanical Models . . . . . . . . . . . . . . . . . . . . . . . . . 804 11 Systems of Differential Equations 812 11.1 Examples of Systems . . . . . . . . . . . . . . . . . . . . . . . . 813 11.2 Fundamental System Methods . . . . . . . . . . . . . . . . . . . 839 11.3 Structure of Linear Systems . . . . . . . . . . . . . . . . . . . . 850 11.4 Matrix Exponential . . . . . . . . . . . . . . . . . . . . . . . . . 865 11.5 Ziebur, Spectral and Eigenanalysis Methods . . . . . . . . . . . 875 11.6 Jordan Form and Eigenanalysis . . . . . . . . . . . . . . . . . . 894 11.7 Nonhomogeneous Linear Systems . . . . . . . . . . . . . . . . . 912 11.8 Second-order Systems . . . . . . . . . . . . . . . . . . . . . . . 923 11.9 Numerical Methods for Systems . . . . . . . . . . . . . . . . . . 942 CONTENTS 12 Series Methods 949 12.1 Review of Calculus Topics . . . . . . . . . . . . . . . . . . . . . 950 12.2 Algebraic Techniques . . . . . . . . . . . . . . . . . . . . . . . . 953 12.3 Power Series Methods . . . . . . . . . . . . . . . . . . . . . . . 959 12.4 Ordinary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . 965 12.5 Regular Singular Points . . . . . . . . . . . . . . . . . . . . . . 968 12.6 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 981 12.7 Legendre Polynomials . . . . . . . . . . . . . . . . . . . . . . . 985 12.8 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996 A Background Topics 1005 A.1 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005 A.2 Graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015 A.3 Explicit and Implicit Answers . . . . . . . . . . . . . . . . . . . 1024 A.4 Numerical and Graphical Answers . . . . . . . . . . . . . . . . 1029 A.5 Implicit Functions . . . . . . . . . . . . . . . . . . . . . . . . . 1041 Index 1045 Bibliography 1061 Paperback and PDF Sources 1065