Modern Differential Equations

1. Introduction to Differential Equations. Introduction. A Graphical Approach to Solutions: Slope Fields and Direction Fields. Summary. Review Exercises. 2. First Order Equations. Separable Equations. First-Order Linear Equations. Substitution Methods and Special Equations. Exact Equations. Theory of First-Order-Equations. Numerical Methods for First-Order Equations. Summary. Review Exercises. Differential Equations at Work. Modeling the Spread of a Disease. Linear Population Model with Harvesting. Logistic Model with Harvesting. Logistic Model with Predation. 3. Applications of First Order Equations. Population Growth and Decay. Newton's Law of Cooling and Related Problems. Free-Falling Bodies. Summary. Review Exercises. Chapter 3 Differential Equations at Work. Mathematics of Finance. Algae Growth. Dialysis. Antibiotic Production. 4. Higher Order Equations. Second-Order Equations: An Introduction. Solutions of Second-Order Linear Homogeneous Equations with Constant Coefficients. Higher Order Equations: An Introduction. Solutions to Higher Order Linear Homogeneous Equations with Constant Coefficients. Introduction to Solving Nonhomogeneous Equations with Constant Coefficients: Method of Undetermined Coefficients. Nonhomogeneous Equations with Constant Coefficients: Variation of Parameters. Cauchy-Euler Equations. Series Solutions of Ordinary Differential Equations. Summary. Review Exercises. Differential Equations at Work. Testing for Diabetes. Modeling the Motion of a Skier. The Schröinger Equation. 5. Applications of Higher Order Equations. Simple Harmonic Motion. Damped Motion. Forced Motion. Other Applications. The Pendulum Problem. Summary. Review Exercises. Differential Equations at Work. Rack-and-Gear Systems. Soft Springs. Hard Springs. Aging Springs. Bodé Plots. 6. Systems of First Order Equations. Introduction. Review of Matrix Algebra and Calculus. Preliminary Definitions and Notation. First-Order Linear Homogeneous Systems with Constant Coefficients. First-Order Linear Nonhomogeneous Systems: Undetermined Coefficients and Variation of Parameters. Phase Portraits. Nonlinear Systems. Numerical Methods. Summary. Review Exercises. Differential Equations at Work. Modeling a Fox Population in Which Rabies is Present. Controlling the Spread of Disease. FitzHugh-Nagumo Model. 7. Applications of First-Order Systems. Mechanical and Electrical Problems with First-Order Linear Systems. Diffusion and Population Problems with First-Order Linear Systems. Nonlinear Systems of Equations. Summary. Review Exercises. Differential Equations at Work. Competing Species. Food Chains. Chemical Reactor. 8. Laplace Transforms. The Laplace Transform: Preliminary Definitions and Notation. Solving Initial-Value Problems with the Laplace Transform. Laplace Transforms of Several Important Functions. The Convolution Theorem. Laplace Transform Methods for Solving Systems. Applications Using Laplace Transforms. Summary. Review Exercises. Differential Equations at Work. The Tautochrone. Vibration Absorbers. Airplane Wing. Free Vibration of a Three-Story Building. Control Systems. 9. Fourier Series. Boundary-Value Problems, Eigenvalue Problems, Sturm-Liouville Problems. Fourier Sine Series and Cosine Series. Fourier Series. Generalized Fourier Series. Summary. Review Exercises. Differential Equations at Work. Free Vibration of a Three-Story Building. Forced Damped Spring-Mass System. Approximations with Fourier Series. 10. Partial Differential Equations. Introduction to Partial Differential Equations and Separation of Variables. The One-Dimensional Heat Equation. The One-Dimensional Wave Equation. Problems in Two Dimensions: Laplace's Equation. Two-Dimensional Problems in a Circular Region. Summary. Review Exercises. Differential Equations at Work. Laplace Transforms. Waves in a Steel Rod. Media Sterilization. Numerical Methods for Solving Partial Differential Equations. Answers to Selected Questions. Index.