Differential Equations: An Introduction to Modern Methods and Applications

1. Introduction. 1.1 Some Basic Mathematical Models Direction Fields. 1.2 Solutions of Some Differential Equations. 1.3 Numerical Approximations: Eulera s Method. 1.4 Classification of Differential Equations. 2. First Order Differential Equations. 2.1 Linear Equations Method of Integrating Factors. 2.2 Separable Equations. 2.3 Modeling with First Order Equations. 2.4 Differences Between Linear and Nonlinear Equations. 2.5 Autonomous Equations and Population Dynamics. 2.6 Exact Equations and Integrating Factors. 2.7 Accuracy of Numerical Methods. 2.8 Improved Euler and Runge Kutta Methods. Summary. Projects. 2.P.1 Harvesting a Renewable Resource. 2.P.2 Designing a Drip Dispenser for a Hydrology Experiment. 2.P.3 A Mathematical Model of a Groundwater Contaminant Source. 2.P.4 Monte Carlo Option Pricing: Pricing Financial Options by Flipping a Coin. 3. Systems of Two First Order Equations. 3.1 Systems of Two Linear Algebraic Equations. 3.2 Systems of Two First Order Linear Differential Equations. 3.3 Homogeneous Linear Systems with Constant Coefficients. 3.4 Complex Eigenvalues. 3.5 Repeated Eigenvalues. 3.6 A Brief Introduction to Nonlinear Systems. 3.7 Numerical Methods for Systems of First Order Equations. Summary. Projects. 3.P.1 Eigenvalue Placement Design of a Satellite Attitude Control System. 3.P.2 Estimating Rate Constants for an Open Two Compartment Model. 3.P.3 The Ray Theory of Wave Propagation. 3.P.4 A Blood Brain Pharmacokinetic Model. 4. Second Order Linear Equations. 4.1 Definitions and Examples. 4.2 Theory of Second Order Linear Homogeneous Equations. 4.3 Linear Homogeneous Equations with Constant Coefficients. 4.4 Characteristic Equations with Complex Roots. 4.5 Mechanical and Electrical Vibrations. 4.6 Nonhomogeneous Equations Method of Undetermined Coefficients. 4.7 Forced Vibrations, Frequency Response, and Resonance. 4.8 Variation of Parameters. Summary. Projects. 4.P.1 A Vibration Insulation Problem. 4.P.2 Linearization of a Nonlinear Mechanical System. 4.P.3 A Spring Mass Event Problem. 4.P.4 Uniformly Distributing Points on a Sphere. 4.P.5 Euler Lagrange Equations. 5. The Laplace Transform. 5.1 Definition of the Laplace Transform. 5.2 Properties of the Laplace Transform. 5.3 The Inverse Laplace Transform. 5.4 Solving Differential Equations with Laplace Transforms. 5.5 Discontinuous Functions and Periodic Functions. 5.6 Differential Equations with Discontinuous Forcing Functions. 5.7 Impulse Functions. 5.8 Convolution Integrals and Their Applications. 5.9 Linear Systems and Feedback Control. Summary. Projects. 5.P.1 An Electric Circuit Problem. 5.P.2 Effects of Pole Locations on Step Responses of Second Order Systems. 5.P.3 The Watt Governor, Feedback Control, and Stability. 6. Systems of First Order Linear Equations. 6.1 Definitions and Examples. 6.2 Basic Theory of First Order Linear Systems. 6.3 Homogeneous Linear Systems with Constant Coefficients. 6.4 Complex Eigenvalues. 6.5 Fundamental Matrices and the Exponential of a Matrix. 6.6 Nonhomogeneous Linear Systems. 6.7 Defective Matrices. Summary. Projects. 6.P.1 A Compartment Model of Heat Flow in a Rod. 6.P.2 Earthquakes and Tall Buildings. 6.P.3 Controlling a Spring Mass System to Equilibrium. 7. Nonlinear Differential Equations and Stability. 7.1 Autonomous Systems and Stability. 7.2 Almost Linear Systems. 7.3 Competing Species. 7.4 Predator Prey Equations. 7.5 Periodic Solutions and Limit Cycles. 7.6 Chaos and Strange Attractors: The Lorenz Equations. Summary. Projects. 7.P.1 Modeling of Epidemics. 7.P.2 Harvesting in a Competitive Environment. 7.P.3 The Rossler System. Appendix A: Matrix Algebra. A.1 Matrices. A.2 Systems of Linear Algebraic Equations, Linear Independence, and Rank. A.3 Determinants and Inverses. A.4 The Eigenvalue Problem. Appendix B: Complex Variables. Answers to Selected Problems. Index.