For introductory courses in Differential Equations. This best-selling text by these well-known authors blends the traditional algebra problem solving skills with the conceptual development and geometric visualization of a modern differential equations course that is essential to science and engineering students. It reflects the new qualitative approach that is altering the learning of elementary differential equations, including the wide availability of scientific computing environments like Maple, Mathematica, and MATLAB. Its focus balances the traditional manual methods with the new computer-based methods that illuminate qualitative phenomena and make accessible a wider range of more realistic applications. Seldom-used topics have been trimmed and new topics added: it starts and ends with discussions of mathematical modeling of real-world phenomena, evident in figures, examples, problems, and applications throughout the text.
1. First Order Differential Equations. Differential Equations and Mathematical Models. Integrals as General and Particular Solutions. Slope Fields and Solution Curves. Separable Equations and Applications. Linear First Order Equations. Substitution Methods and Exact Equations. 2. Mathematical Models and Numerical Methods. Population Models. Equilibrium Solutions and Stability. Acceleration-Velocity Models. Numerical Approximation: Eulers Method. A Closer Look at the Euler Method, and Improvements. The Runge-Kutta Method. 3. Linear Equations of Higher Order. Introduction: Second-Order Linear Equations. General Solutions of Linear Equations. Homogeneous Equations with Constant Coefficients. Mechanical Vibrations. Nonhomogeneous Equations and Undetermined Coefficients. Forced Oscillations and Resonance. Electrical Circuits. Endpoint Problems and Eigenvalues. 4. Introduction to Systems of Differential Equations. First-Order Systems and Applications. The Method of Elimination. Numerical Methods for Systems. 5. Linear Systems of Differential Equations. Linear Systems and Matrices. The Eigenvalue Method for Homogeneous Systems. Second Order Systems and Mechanical Applications. Multiple Eigenvalue Solutions. Matrix Exponentials and Linear Systems. Nonhomogenous Linear Systems. 6. Nonlinear Systems and Phenomena. Stability and the Phase Plane. Linear and Almost Linear Systems. Ecological Models: Predators and Competitors. Nonlinear Mechanical Systems. Chaos in Dynamical Systems. 7. Laplace Transform Methods. Laplace Transforms and Inverse Transforms. Transformation of Initial Value Problems. Translation and Partial Fractions. Derivatives, Integrals, and Products of Transforms. Periodic and Piecewise Continuous Forcing Functions. Impulses and Delta Functions. 8. Power Series Methods. Introduction and Review of Power Series. Series Solutions Near Ordinary Points. Regular Singular Points. Method of Frobenius: The Exceptional Cases. Bessels Equation. Applications of Bessel Functions. 9. Fourier Series Methods. Periodic Functions and Trigonometric Series. General Fourier Series and Convergence. Even-Odd Functions and Termwise Differentiation. Applications of Fourier Series. Heat Conduction and Separation of Variables. Vibrating Strings and the One-Dimensional Wave Equation. Steady-State Temperature and Laplaces Equation. 10. Eigenvalues and Boundary Value Problems. Sturm-Liouville Problems and Eigenfunction Expansions. Applications of Eigenfunction Series. Steady Periodic Solutions and Natural Frequencies. Applications of Bessel Functions. Higher-Dimensional Phenomena. References. Appendix: Existence and Uniqueness of Solutions. Answers to Selected Problems. Index.