Title: Differential Equations: Theory,Technique and Practice with Boundary Value Problems
Abstract: What Is a Differential Equation? Introductory Remarks A Taste of Ordinary Differential Equations The Nature of Solutions Separable Equations First-Order Linear Equations Exact Equations Orthogonal Trajectories and Families of Curves Homogeneous Equations Integrating Factors Reduction of Order The Hanging Chain and Pursuit Curves Electrical Circuits Anatomy of an Application Problems for Review and Discovery Second-Order Linear Equations Second-Order Linear Equations with Constant Coefficients The Method of Undetermined Coefficients The Method of Variation of Parameters The Use of a Known Solution to Find Another Vibrations and Oscillations Newton's Law of Gravitation and Kepler's Laws Higher-Order Equations Historical Note: Euler Anatomy of an Application Problems for Review and Discovery Power Series Solutions and Special Functions Introduction and Review of Power Series Series Solutions of First-Order Equations Second-Order Linear Equations: Ordinary Points Regular Singular Points More on Regular Singular Points Gauss's Hypergeometric Equation Historical Note: Gauss Historical Note: Abel Anatomy of an Application Problems for Review and Discovery Numerical Methods Introductory Remarks The Method of Euler The Error Term An Improved Euler Method The Runge-Kutta Method Anatomy of an Application Problems for Review and Discovery Fourier Series: Basic Concepts Fourier Coefficients Some Remarks about Convergence Even and Odd Functions: Cosine and Sine Series Fourier Series on Arbitrary Intervals Orthogonal Functions Historical Note: Riemann Anatomy of an Application Problems for Review and Discovery Sturm-Liouville Problems and Boundary Value Problems What Is a Sturm-Liouville Problem? Analyzing a Sturm-Liouville Problem Applications of the Sturm-Liouville Theory Singular Sturm-Liouville Anatomy of an Application Problems for Review and Discovery Partial Differential Equations and Boundary Value Problems Introduction and Historical Remarks Eigenvalues, Eigenfunctions, and the Vibrating String The Heat Equation The Dirichlet Problem for a Disc Historical Note: Fourier Historical Note: Dirichlet Problems for Review and Discovery Anatomy of an Application Laplace Transforms Introduction Applications to Differential Equations Derivatives and Integrals of Laplace Transforms Convolutions The Unit Step and Impulse Functions Historical Note: Laplace Anatomy of an Application Problems for Review and Discovery Systems of First-Order Equations Introductory Remarks Linear Systems Homogeneous Linear Systems with Constant Coefficients Nonlinear Systems: Volterra's Predator-Prey Equations Anatomy of an Application Problems for Review and Discovery The Nonlinear Theory Some Motivating Examples Specializing Down Types of Critical Points: Stability Critical Points and Stability for Linear Systems Stability by Liapunov's Direct Method Simple Critical Points of Nonlinear Systems Nonlinear Mechanics: Conservative Systems Periodic Solutions: The Poincare-Bendixson Theorem Historical Note: Poincare Anatomy of an Application Problems for Review and Discovery Appendix: Review of Linear Algebra
Publication Year: 2015
Publication Date: 2015-10-16
Language: en
Type: book
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