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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 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 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 Method for Partial Fractions 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 Eigenanalysis, Spectral and CHZ 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 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