For one- or two-semester, sophomore-level courses in Differential Equations. The text introduces first order systems in the first chapter and uses them as the main focus for the treatment of ordinary differential equations covered in the first seven chapters. This allows an early and unified introduction of graphical and numerical methods along with analytical methods. The graphical and numerical theme is sustained in the chapters on partial differential equations and Fourier series.

Table of Contents:
(NOTE: Most chapters conclude with Supplementary Exercises.) 1. Introduction. What Is a Differential Equation? Applications of Differential Equations. Approaches to Solving Differential Equations. Reduction to First-Order Systems. 2. The First-Order Equation y' = f (x, y) . The Graphical Viewpoint (Direction Fields). Numerical Methods. Analytic Methods of Solution. Autonomous Equations and Critical Points. The Dependence of Solutions on Initial Conditions. 3. Introduction to First-Order Systems. Solutions to First-Order Systems. Orbit Crossing and Periodic Solutions. Numerical Approximations. Some Qualitative Behavior. The Pendulum. Limit Cycles. 4. Higher-Order Linear Equations. Introduction. A Strategy for Solving Linear Homogeneous Equations. Linear Homogeneous Equations with Constant Coefficients. NonHomogeneous Equations. Vibration. 5. First-Order Systems: Linear Methods. Matrices, Independence, and Eigenvectors. Solving 2 x 2 Linear Systems. The Matrix Exponential Function. Qualitative Behavior of Linear Systems. A Couple System of Masses and Springs. Linearization of 2 x 2 Systems. 6. Series Methods and Famous Functions. A Power Series Method. Famous Functions. Regular Singular Points and the Method of Frobenius. The Exceptional Cases. 7. Bifurcations and Chaos. Bifurcation. Flows. A Basic Theorem. Some Simple Attractors. The Periodically Driven Pendulum. Chaos. 8. The Laplace Transform. Introduction. Transforms of Basic Functions. Solving Linear Homogeneous Equations. NonHomogeneous Equations and the Convolution. Discontinuous and Impulsive Forcing Functions. Laplace Transforms and Systems. Poles and Qualitative Behavior. 9. Partial Differential Equations and Fourier Series. Some General Remarks. The Heat Equation, Wave Equation, and Laplace's Equation. The Heat Equation and Initial Condition. Vector Spaces and Operators. The Heat Equation Revisited. Periodic Functions and Fourier Series. The One-Dimensional Wave Equation. The Convergence of Series. The Two-Dimensional Wave Equation. Laplace's Equation. 10. The Finite Differences Method. Finite Differences Approximations. An Example. The Heat and Wave Equations. A Backward Method for the Heat Equation. Variable Coefficient and NonLinear Examples. Laplace's Equation. Stability. APPENDICES. Appendix A: Linear Systems, Matrices, and Determinants. Solution of Systems of Equations. Matrices, Inverses, and Determinants. Appendix B: The Two-Variable Taylor Theorem. Appendix C: The Existence and Uniqueness Theorem. Appendix D: Mathematica, Maple, and MatLab. Graphing. Field&Solution. ERGraphical. ERNumerical. 2x2Systems. 2x2Numerical. Fourier Series. Appendix E: Answers and Hints to Odd-Numbered Exercises. Bibliography. Index.