A Modern Introduction to Differential Equations

Contents
Preface
	Philosophy
	Use of technology
	Pedagogical features and writing style
	Preface
	New to the Third Edition
	Supplements
Acknowledgments
1 Introduction to differential equations
	Introduction
	1.1 Basic terminology
		1.1.1 Ordinary and partial differential equations
			Ordinary differential equations
			Partial differential equations
			The order of an ordinary differential equation
			A general form for an ordinary differential equation
			Linear and nonlinear ordinary differential equations
		1.1.2 Systems of ordinary differential equations
	1.2 Solutions of differential equations
		1.2.1 Basic notions
			Implicit solutions
		1.2.2 Families of solutions I
	1.3 Initial-value problems and boundary-value problems
		1.3.1 An integral form of an IVP solution
		1.3.2 Families of solutions II
			Boundary-value problems
			General solutions
		1.3.3 Solutions of systems of ODEs
	Summary
2 First-order differential equations
	Introduction
	2.1 Separable equations
	2.2 Linear equations
		2.2.1 The superposition principle
		2.2.2 Variation of parameters and the integrating factor
	2.3 Compartment problems
	2.4 Slope fields
		2.4.1 Autonomous and nonautonomous equations
	2.5 Phase lines and phase portraits
		2.5.1 The logistic equation
	2.6 Equilibrium points: sinks, sources, and nodes
		2.6.1 A test for equilibrium points
	2.7 Bifurcations
		2.7.1 Basic concepts
		2.7.2 Application to differential equations
	2.8 Existence and uniqueness of solutions
		2.8.1 An Existence and Uniqueness Theorem
		2.8.2 A sketch of a proof of the Existence and Uniqueness Theorem
	Summary
3 The numerical approximation of solutions
	Introduction
	3.1 Euler's method
		3.1.1 Stiff differential equations
	3.2 The improved Euler method
	3.3 More sophisticated numerical methods: Runge-Kutta and others
	Summary
4 Second- and higher-order equations
	Introduction
	4.1 Homogeneous second-order linear equations with constant coefficients
		4.1.1 The characteristic equation and eigenvalues
		4.1.2 Real but unequal roots
		4.1.3 Real but equal roots
		4.1.4 Complex conjugate roots
		4.1.5 The amplitude-phase angle form of a solution
		4.1.6 Summary
	4.2 Nonhomogeneous second-order linear equations with constant coefficients
		4.2.1 The structure of solutions
	4.3 The method of undetermined coefficients
	4.4 Variation of parameters
	4.5 Higher-order linear equations with constant coefficients
	4.6 Existence and uniqueness
		4.6.1 An Existence and Uniqueness Theorem
		4.6.2 Many solutions
		4.6.3 No solution
		4.6.4 Exactly one solution
	Summary
5 The Laplace transform
	Introduction
	5.1 The Laplace transform of some important functions
	5.2 The inverse transform and the convolution
		5.2.1 The inverse Laplace transform
		5.2.2 The convolution
		5.2.3 Integral equations and integro-differential equations
		5.2.4 The Laplace transform and technology
	5.3 Transforms of discontinuous functions
		5.3.1 The Heaviside (unit step) function
	5.4 Transforms of impulse functions-the Dirac delta function
	5.5 Transforms of systems of linear differential equations
	5.6 Laplace transforms of linear differential equations with variable coefficients
	Summary
6 Systems of linear differential equations
	Introduction
	6.1 Higher-order equations and their equivalent systems
		6.1.1 Conversion technique I: converting a higher-order equation into a system
		6.1.2 Conversion technique II: converting a system into a higher-order equation
	6.2 Existence and uniqueness
		6.2.1 An Existence and Uniqueness Theorem
		6.2.2 Many solutions
		6.2.3 No solution
		6.2.4 Exactly one solution
	6.3 Numerical solutions of systems
		6.3.1 Euler's method applied to systems
		6.3.2 The fourth-order Runge-Kutta method for systems
	6.4 The geometry of autonomous systems
		6.4.1 Phase portraits for systems of equations
		6.4.2 Equilibrium points
		6.4.3 Three-dimensional systems
	6.5 Systems and matrices
		6.5.1 Matrices and vectors
		6.5.2 The matrix representation of a linear system
		6.5.3 Some matrix algebra
	6.6 Two-dimensional systems of first-order linear equations
		6.6.1 Eigenvalues and eigenvectors
		6.6.2 Geometric interpretation of eigenvectors
		6.6.3 The general problem
		6.6.4 The geometric behavior of solutions
	6.7 The stability of homogeneous linear systems: unequal real eigenvalues
		6.7.1 Unequal real eigenvalues
		6.7.2 The impossibility of dependent eigenvectors
		6.7.3 Unequal positive eigenvalues
		6.7.4 Unequal negative eigenvalues
		6.7.5 Unequal eigenvalues with opposite signs
		6.7.6 Unequal eigenvalues, one eigenvalue equal to zero
	6.8 The stability of homogeneous linear systems: equal real eigenvalues
		6.8.1 Equal nonzero eigenvalues, two independent eigenvectors
		6.8.2 Equal nonzero eigenvalues, only one independent eigenvector
		6.8.3 Both eigenvalues zero
	6.9 The stability of homogeneous linear systems: complex eigenvalues
		6.9.1 Complex eigenvalues and complex eigenvectors
	6.10 Nonhomogeneous systems
		6.10.1 The general solution
		6.10.2 The method of undetermined coefficients
	6.11 Spring-mass problems
		6.11.1 Simple harmonic motion
		6.11.2 Analysis
		6.11.3 Another view-solution curves
		6.11.4 Free damped motion
		6.11.5 Different kinds of damping
		6.11.6 Forced motion
		6.11.7 Resonance
		6.11.8 An analogy
	6.12 Generalizations: the n xn case (n >=3)
		6.12.1 Matrix representation
		6.12.2 Eigenvalues and eigenvectors
		6.12.3 Linear independence and linear dependence
		6.12.4 Nonhomogeneous systems
		6.12.5 Generalization to n xn systems
	Summary
7 Systems of nonlinear differential equations
	Introduction
	7.1 Equilibria of nonlinear systems
	7.2 Linear approximation at equilibrium points
		7.2.1 Almost linear systems
	7.3 The Hartman-Grobman theorem
	7.4 Two important nonlinear systems
		7.4.1 A predator-prey model: the Lotka-Volterra equations
		7.4.2 A qualitative analysis of the Lotka-Volterra equations
		7.4.3 Other graphical representations
		7.4.4 The undamped pendulum
	7.5 Bifurcations
	7.6 Limit cycles and the Hopf bifurcation
		7.6.1 Limit cycles
		7.6.2 The Hopf bifurcation
	Summary
A Some calculus concepts and results
	A.1 Local linearity: the tangent line approximation
	A.2 The chain rule
	A.3 The Taylor polynomial/Taylor series
	A.4 The fundamental theorem of calculus
	A.5 Partial fractions
	A.6 Improper integrals
	A.7 Functions of several variables/partial derivatives
	A.8 The tangent plane: the Taylor expansion of F(x,y)
B Vectors and matrices
	B.1 Vectors and vector algebra; polar coordinates
	B.2 Matrices and basic matrix algebra
	B.3 Linear transformations and matrix multiplication
	B.4 Eigenvalues and eigenvectors
C Complex numbers
	C.1 Complex numbers: the algebraic view
	C.2 Complex numbers: the geometric view
	C.3 The quadratic formula
	C.4 Euler's formula
D Series solutions of differential equations
	D.1 Power series solutions of first-order equations
	D.2 Series solutions of second-order linear equations: ordinary points
	D.3 Regular singular points: the method of Frobenius
	D.4 The point at infinity
	D.5 Some additional special differential equations
Answers and hints to odd-numbered exercises
Index