دانلود کتاب Differential Equations: From Calculus to Dynamical Systems

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Cover
Title Page
Copyright Page
Table of Contents
Preface
1 Introduction to Differential Equations
1.1 Basic Terminology
1.1.1 Ordinary vs. Partial Differential Equations
1.1.2 Independent Variables, Dependent Variables, and Parameters
1.1.3 Order of a Differential Equation
1.1.4 What is a Solution?
1.1.5 Systems of Differential Equations
1.2 Families of Solutions, Initial-Value Problems
1.3 Modeling with Differential Equations
2 First-order Differential Equations
2.1 Separable First-order Equations
2.1.1 Application 1: Population Growth
2.1.2 Application 2: Newton’s Law of Cooling
2.2 Graphical Methods, the Slope Field
2.2.1 Using Graphical Methods to Visualize Solutions
2.3 Linear First-order Differential Equations
2.3.1 Application: Single-compartment Mixing Problem
2.4 Existence and Uniqueness of Solutions
2.5 More Analytic Methods for Nonlinear First-order Equations
2.5.1 Exact Differential Equations
2.5.2 Bernoulli Equations
2.5.3 Using Symmetries of the Slope Field
2.6 Numerical Methods
2.6.1 Euler’s Method
2.6.2 Improved Euler Method
2.6.3 Fourth-order Runge-Kutta Method
2.7 Autonomous Equations, the Phase Line
2.7.1 Stability—Sinks, Sources, and Nodes
Bifurcation in Equations with Parameters
3 Second-order Differential Equations
3.1 General Theory of Homogeneous Linear Equations
3.2 Homogeneous Linear Equations with Constant Coefficients
3.2.1 Second-order Equation with Constant Coefficients
3.2.2 Equations of Order Greater Than Two
3.3 The Spring-mass Equation
3.3.1 Derivation of the Spring-mass Equation
3.3.2 The Unforced Spring-mass System
3.4 Nonhomogeneous Linear Equations
3.4.1 Method of Undetermined Coefficients
3.4.2 Variation of Parameters
3.5 The Forced Spring-mass System
Beats and Resonance
3.6 Linear Second-order Equations with Nonconstant Coefficients
3.6.1 The Cauchy-Euler Equation
3.6.2 Series Solutions
3.7 Autonomous Second-order Differential Equations
3.7.1 Numerical Methods
3.7.2 Autonomous Equations and the Phase Plane
4 Linear Systems of First-order Differential Equations
4.1 Introduction to Systems
4.1.1 Writing Differential Equations as a First-order System
4.1.2 Linear Systems
4.2 Matrix Algebra
4.3 Eigenvalues and Eigenvectors
4.4 Analytic Solutions of the Linear System ⃗