Table of Contents Chapter 1 Basic Concepts      1.1 Simple Differential Equations and Explicit Solutions      1.2 Graphical Solutions Using Calculus      1.3 Slope Fields and Isoclines      1.4 Functions and Power Series Expansions Chapter 2 Autonomous Differential Equations      2.1 Autonomous Equations      2.2 Simple Models      2.3 The Logistic Equation      2.4 Existence and Uniqueness of Solutions, and Words of Caution      2.5 Qualitative Behavior of Solutions Using Phase Lines      2.6 Bifurcation Diagrams Chapter 3 First Order Differential Equations—Qualitative and Quantitative Aspects      3.1 Graphical Solutions Using Calculus      3.2 Symmetry of Slope Fields      3.3 Numerical Solutions and Chaos      3.4 Comparing Solutions of Differential Equations      3.5 Finding Power Series Solutions Chapter 4 Models and Applications Leading to New Techniques      4.1 Solving Separable Differential Equations      4.2 Solving Differential Equations with Homogeneous Coefficients      4.3 Models - Deriving Differential Equations From Data      4.4 Models - Objects in Motion      4.5 Application - Orthogonal Trajectories      4.6 Piecing Together Differential Equations Chapter 5 First Order Linear Differential Equations and Models      5.1 Solving Linear Differential Equations      5.2 Models That Use Linear Equations      5.3 Models That Use Bernoulli's Equation Chapter 6 Interplay Between First Order Systems and Second Order Equations      6.1 Simple Models      6.2 How First Order Systems and Second Order Equations Are Related      6.3 Second Order Linear Differential Equations with Constant Coefficients      6.4 Modeling Physical Situations      6.5 Interpreting the Phase Plane      6.6 How Explicit Solutions Are Related to Orbits      6.7 The Motion of a Nonlinear Pendulum Chapter 7 Second Order Linear Differential Equations with Forcing Functions      7.1 The General Solution      7.2 Finding Solutions by the Method of Undetermined Coefficients      7.3 Applications and Models Chapter 8 Second Order Linear Differential Equations—Qualitative and Quantitative Aspects      8.1 Qualitative Behavior of Solutions      8.2 Finding Solutions by Reduction of Order      8.3 Finding Solutions by Variation of Parameters      8.4 The Importance of Linear Independence and Dependence      8.5 Solving Cauchy-Euler Equations      8.6 Boundary Value Problems and the Shooting Method      8.7 Solving Higher Order Homogeneous Differential Equations      8.8 Solving Higher Order Nonhomogeneous Differential Equations Chapter 9 Linear Autonomous Systems      9.1 Solving Linear Autonomous Systems      9.2 Classification of Solutions via Stability      9.3 When Do Straight-Line Orbits Exist?      9.4 Qualitative Behavior Using Nullclines      9.5 Matrix Formulation of Solutions      9.6 Compartmental Models Chapter 10 Nonlinear Autonomous Systems      10.1 Introduction to Nonlinear Autonomous Systems      10.2 Qualitative Behavior Using Nullclines Analysis      10.3 Qualitative Behavior Using Linearization      10.4 Models Involving Nonlinear Autonomous Equations      10.5 Bungee Jumping      10.6 Linear Versus Nonlinear Differential Equations      10.7 Autonomous Versus Nonautonomous Differential Equations Chapter 11 Using Laplace Transforms      11.1 Motivation      11.2 Constructing New Laplace Transforms from Old      11.3 The Inverse Laplace Transform and the Convolution Theorem      11.4 Functions That Jump      11.5 Models Involving First Order Linear Differential Equations      11.6 Models Involving Higher Order Linear Differential Equations      11.7 Applications to Systems of Linear Differential Equations      11.8 When Do Laplace Transforms Exist? Chapter 12 Using Power Series      12.1 Solutions Using Taylor Series      12.2 Solutions Using Power Series      12.3 What To Do When Power Series Fail      12.4 Solutions Using The Method of Frobenius Appendices      A.1 Background Material      A.2 Partial Fractions      A.3 Infinite Series, Power Series, and Taylor Series      A.4 Complex Numbers      A.5 Elementary Matrix Operations      A.6 Least Squares Approximation      A.7 Proofs of the Oscillation Theorems