Differential Equations / Edition 1by John Polking
Pub. Date: 01/28/2002
Publisher: Prentice Hall Professional Technical Reference
This text strikes a balance between the traditional and the modern. It covers all of the traditional material and more. It offers flexibility of use that will allow faculty at a variety of institutions to use the text. Includes a large number of graphics that display the ideas in ODEs; modeling and applications; and projects, which provide students with the opportunity to bring together much of what they have learned, including analytical, computational, and interpretative skills. Ideal for readers interested in learning about differential equations.
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Table of Contents1. Introduction to Differential Equations.
Differential Equation Models. The Derivative. Integration.
2. First Order Equations.
Differential Equations and Solutions. Solutions to Separable Equations. Models of Motion. Linear Equations. Mixing Problems. Exact Equations. Existence and Uniqueness of Solutions. Dependence of Solutions on Initial Conditions. Autonomous Equations and Stability.
3. Modeling and Applications.
Modeling. Modeling Population Growth. Personal Finance. Electrical Circuits. Personal Finance.
4. Second Order Equations.
Two Applications. Terminology and Notation. Second Order Equations with Constant Coefficients. Vibrating Springs ---Analysis. Inhomogeneous Systems and Equations. Method of Undetermined Coefficients. Variation of Parameters. Forced Harmonic Motion. Electric Circuits.
5. Laplace Transforms.
The Definition of Laplace Transform. Basic Properties. The Inverse Laplace Transform. Using the Laplace Transform to Solve Differential Equations. Discontinuous Forcing Terms. The Delta Function. Convolutions. Summary.
6. Numerical Methods.
Euler's Method. Runge-Kutta Methods. Numerical Error Comparisons. Other Solvers. A Cautionary Tale.
7. Matrix Algebra.
Vectors and Matrices. The Geometry of Systems of Linear Equations. Solving Systems of Equations. Properties of Solution Spaces. Subspaces. Determinants.
8. Linear Systems of Differential Equations.
Definition of a Linear System. Modeling with Systems. Geometrical Interpretation. Qualitative Analysis. Homogenous Systems and Equations.
9. Linear Systems with Constant Coefficients.
Overview of the Technique. Planar Systems. Phase Plane Portraits. Higher Dimensional Systems. The Exponential of a Matrix. Qualitative Analysis of Linear Systems. Higher Order Linear Equations. Computing the Exponential of a Matrix. Inhomogeneous Systems and Equations. Variation of Parameters for Systems.
10. Nonlinear Systems.
The Linearization of a Nonlinear System. Long Term Behavior of Solutions. Invariant Sets and the Use of Nullclines. Long Term Behavior of Solutions to Planar Systems. Conserved Quantities. Nonlinear Mechanics. The Method of Lyapunov. Predator-Prey Systems. Immune Response to Infectious Disease.
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