Introductory Differential Equations

Preface
	Technology
	Applications
	Style
	Features
	Pedagogical Features
		Examples
		\"Think about it!\"
		Technology
		Exercises
		Chapter Summary and Review Exercises
		Figures
		Historical Material
	Content
Chapter 1: Introduction to Differential Equations
	1.1 Introduction to Differential Equations: Vocabulary
	Exercises 1.1
	1.2 A Graphical Approach to Solutions: Slope Fields and Direction Fields
		Systems of Ordinary Differential Equations and Direction Fields
		Relationship Between Systems of First Order and Higher Order Equations
	Exercises 1.2
	Chapter 1 Summary: Essential Concepts and Formulas
	Chapter 1 Review Exercises
Chapter 2: First Order Ordinary Differential Equations
	2.1 Introduction to First Order Equations
		Some Differences Between Linear Equations and Nonlinear Equations
	Exercises 2.1
	2.2 Separable Equations
		Equilibrium Solutions of dy/dt=f(y)
	Exercises 2.2
	2.3 First Order Linear Equations: Undetermined Coefficients
		2.3.1 General Solution of a First Order Linear Equation
		2.3.2 Method of Undetermined Coefficients
			2.3.2.1 Outline of the Method of Undetermined Coefficients to Solve  y\'+ky=q(t) Where k Is Constant and q(t) Is a Linear Combination of the Functions in (2.5)
			2.3.2.2 Determining the Form of yp(t) (Step 2)
	Exercises 2.3
	2.4 First Order Linear Equations: Integrating Factor
	Exercises 2.4
	2.5 Exact Differential Equations
	Exercises 2.5
	2.6 Substitution Methods and Special Equations
	Exercises 2.6
	2.7 Numerical Methods for First Order Equations
		Euler\'s Method
		Improved Euler\'s Method
		Errors
		Runge-Kutta Method
		Computer Assisted Solutions using Commercial Software
	Exercises 2.7
	Chapter 2 Summary: Essential Concepts and Formulas
	Chapter 2 Review Exercises
	Differential Equations at Work
		A. Modeling the Spread of a Disease
		B. Linear Population Model with Harvesting
		C. Logistic Model with Harvesting
		D. Logistic Model with Predation
		References
Chapter 3: Applications of First Order Differential Equations
	3.1 Population Growth and Decay
		The Logistic Equation
		Population Model with a Threshold
	Exercises 3.1
	3.2 Newton\'s Law of Cooling and Related Problems
		Newton\'s Law of Cooling
		Mixture Problems
	Exercises 3.2
	3.3 Free-Falling Bodies
	Exercises 3.3
	Chapter 3 Summary: Essential Concepts and Formulas
	Chapter 3 Review Exercises
	Differential Equations at Work
		A. Mathematics of Finance
		B. Algae Growth
		C. Dialysis
		D. Antibiotic Production
Chapter 4: Higher Order Equations
	4.1 Second Order Equations: An Introduction
		The Second Order Linear Homogeneous Equation with Constant Coefficients
		The General Case
		Reduction of Order
	Exercises 4.1
	4.2 Solutions of Second Order Linear Homogeneous Equations with Constant Coefficients
		Two Distinct Real Roots
		Complex Conjugate Roots
			Solving Second Order Equations with Constant Coefficients
	Exercises 4.2
	4.3 Solving Second Order Linear Equations: Undetermined Coefficients
		Basic Theory
		Method of Undetermined Coefficients
			4.3.0.1 Outline of the Method of Undetermined Coefficients to Solve
			4.3.0.2 Determining the Form of yp(t) (Step 2)
	Exercises 4.3
	4.4 Solving Second Order Linear Equations: Variation of Parameters
		4.4.0.1 Summary of Variation of Parameters for Second Order Equations
	Exercises 4.4
		Green\'s Functions
	4.5 Solving Higher Order Linear Homogeneous Equations
		Basic Theory
		Constant Coefficients
			4.5.0.1 Distinct Real Roots
			4.5.0.2 Repeated Real Roots
			4.5.0.3 Complex Conjugate Roots
			4.5.0.4 Determining a General Solution of a Higher Order Equation
	Exercises 4.5
	4.6 Solving Higher Order Linear Equations: Undetermined Coefficients and Variation of Parameters
		General Solution of a Nonhomogeneous Equation
		Undetermined Coefficients
		Variation of Parameters
	Exercises 4.6
	4.7 Cauchy-Euler Equations
		Second Order Cauchy-Euler Equations
		Nonhomogeneous Cauchy-Euler Equations
		Higher Order Cauchy-Euler Equations
	Exercises 4.7
	4.8 Power Series Solutions of Ordinary Differential Equations
		Series Solutions about Ordinary Points
		Legendre\'s Equation
	Exercises 4.8
	4.9 Series Solutions of Ordinary Differential Equations
		Regular and Irregular Points and the Method of Frobenius
		The Gamma Function
		Bessel\'s Equation
	Exercises 4.9
	Chapter 4 Summary: Essential Concepts and Formulas
	Chapter 4 Review Exercises
	Differential Equations at Work
		A. Testing for Diabetes
		B. Modeling the Motion of a Skier
		C. The Schrödinger Equation
Chapter 5: Applications of Higher Order Differential Equations
	5.1 Simple Harmonic Motion
	Exercises 5.1
	5.2 Damped Motion
	Exercises 5.2
	5.3 Forced Motion
	Exercises 5.3
	5.4 Other Applications
		L-R-C Circuits
		Deflection of a Beam
	Exercises 5.4
	5.5 The Pendulum Problem
	Exercises 5.5
	Chapter 5 Summary: Essential Concepts and Formulas
	Chapter 5 Review Exercises
	Differential Equations at Work
		A. Rack-and-Gear Systems
		B. Soft, Hard, and Aging Springs
		C. Bodé Plots
		D. The Catenary
		E. The Wave Equation on a Circular Plate
		F. Duffing\'s Equation
		G. Suspending an Object from a Cable
		H. Can Resonance Impact Machinery?
		I. Inventory Management
		J. Heat Transfer
Chapter 6: Systems of Differential Equations
	6.1 Introduction
	Exercises 6.1
	6.2 Review of Matrix Algebra and Calculus
		Basic Operations
		Determinants and Inverses
		Eigenvalues and Eigenvectors
		Matrix Calculus
	Exercises 6.2
	6.3 An Introduction to Linear Systems
	Exercises 6.3
	6.4 First Order Linear Homogeneous Systems with Constant Coefficients
		Distinct Real Eigenvalues
		Complex Conjugate Eigenvalues
		Alternate Method for Solving Initial Value Problems
		Repeated Eigenvalues
	Exercises 6.4
	6.5 First Order Linear Nonhomogeneous Systems: Undetermined Coefficients and Variation of Parameters
		Undetermined Coefficients
		Variation of Parameters
	Exercises 6.5
	6.6 Phase Portraits
		Real Distinct Eigenvalues
		Repeated Eigenvalues
		Complex Conjugate Eigenvalues
		Stability
	Exercises 6.6
	6.7 Nonlinear Systems
		6.7.0.1 Classification of Equilibrium Points of a Nonlinear System
	Exercises 6.7
	6.8 Numerical Methods
		Euler\'s Method
		Runge-Kutta Method
		Computer Algebra Systems and Other Software
	Exercises 6.8
	Chapter 6 Summary: Essential Concepts and Formulas
	Chapter 6 Review Exercises
	Differential Equations at Work
		A. Modeling a Fox Population in which Rabies is Present
		B. Controlling the Spread of a Disease
		C. FitzHugh-Nagumo Model
		D. An Agricultural Model
		E. Modeling the Spread of Dengue in Indonesia
Chapter 7: Applications of Systems of Ordinary Differential Equations
	7.1 Mechanical and Electrical Problems with First Order Linear Systems
		L-R-C Circuits with Loops
		L-R-C Circuit with One Loop
		L-R-C Circuit with Two Loops
		Spring-Mass Systems
	Exercises 7.1
	7.2 Diffusion and Population Problems with First Order Linear Systems
		Diffusion Through a Membrane
		Mixture Problems
		Population Problems
	Exercises 7.2
	7.3 Nonlinear Systems of Equations
		Biological Systems: Predator-Prey Interaction
		Physical Systems: Variable Damping
	Exercises 7.3
	Chapter 7 Summary: Essential Concepts and Formulas
	Chapter 7 Review Exercises
	Differential Equations at Work
		A. Competing Species
		B. Food Chains
		C. Chemical Reactor
		D. Food Chains in a Chemostat
			7.3.0.1 Simple Food Chain in a Chemostat
		E. The Rössler System and Attractor
		F. Cell Dynamics in Colon Cancer
Chapter 8: Introduction to the Laplace Transform
	8.1 The Laplace Transform: Preliminary Definitions and Notation
		Definition of the Laplace Transform
		Exponential Order, Jump Discontinuities and Piecewise-Continuous Functions
		Properties of the Laplace Transform
	Exercises 8.1
	8.2 The Inverse Laplace Transform
	Exercises 8.2
	8.3 Solving Initial-Value Problems with the Laplace Transform
	Exercises 8.3
	8.4 Laplace Transforms of Several Important Functions
		8.4.1 Piecewise Defined Functions: The Unit Step Function
		8.4.2 Periodic Functions
		8.4.3 Impulse Functions, The Delta Function
	Exercises 8.4
	8.5 The Convolution Theorem
		8.5.1 The Convolution Theorem
		8.5.2 Integral and Integrodifferential Equations
	Exercises 8.5
	8.6 Laplace Transform Methods for Solving Systems
	Exercises 8.6
	8.7 Some Applications Using Laplace Transforms
		8.7.1 L-R-C Circuits Revisited
		8.7.2 Delay Differential Equations
		8.7.3 Coupled Spring-Mass Systems
		8.7.4 The Double Pendulum
	Exercises 8.7
	Chapter 8 Summary: Essential Concepts and Formulas
	Chapter 8 Review Exercises
	Differential Equations at Work
		A. The Tautochrone
		B. Vibration Absorbers
		C. Airplane Wing
		D. Free Vibration of a Three-Story Building
		E. Control Systems
Answers to Selected Exercises
	Exercises 1.1
	Exercises 1.2
	Chapter 1 Review Exercises
	Exercises 2.1
	Exercises 2.2
	Exercises 2.3
	Exercises 2.4
	Exercises 2.5
	Exercises 2.6
	Exercises 2.7
	Chapter 2 Review Exercises
	Exercises 3.1
	Exercises 3.2
	Exercises 3.3
	Chapter 3 Review Exercises
	Exercises 4.1
	Exercises 4.2
	Exercises 4.3
	Exercises 4.4
	Exercises 4.5
	Exercises 4.6
	Exercises 4.7
	Exercises 4.8
	Exercises 4.9
	Chapter 4 Review Exercises
	Exercises 5.1
	Exercises 5.2
	Exercises 5.3
	Exercises 5.4
	Exercises 5.5
	Chapter 5 Review Exercises
	Exercises 6.1
	Exercises 6.2
	Exercises 6.3
	Exercises 6.4
	Exercises 6.5
	Exercises 6.6
	Exercises 6.7
	Exercises 6.8
	Chapter 6 Review Exercises
	Exercises 7.1
	Exercises 7.2
	Exercises 7.3
	Chapter 7 Review Exercises
	Exercises 8.1
	Exercises 8.2
	Exercises 8.3
	Exercises 8.4
	Exercises 8.5
	Exercises 8.6
	Exercises 8.7
	Chapter 8 Review Exercises
Bibliography
Index