includes index

Chapter 1 Introduction 1
1.1 Some Basic Mathematical Models; Direction Fields;
1.2 Solutions of Some Differential Equations;
1.3 Classification of Differential Equations;
1.4 Historical Remarks
Chapter 2 First Order Differential Equations;
2.1 ;Linear Equations; Method of Integrating Factors;
2.2;Separable Equations;
2.3 Modeling with First Order Equations;
2.4 Differences Between Linear and Nonlinear Equations
2.5 Autonomous Equations and Population Dynamics
2.6 Exact Equations and Integrating Factors;
2.7 Numerical Approximations: Euler's Method;
2.8 The Existence and Uniqueness Theorem
2.9 First Order Difference Equations;
Chapter 3 Second Order Linear Equations 135
3.1 Homogeneous Equations with Constant Coefficients;
3.2 Fundamental Solutions of Linear Homogeneous Equations; The Wronskian;
3.3 Complex Roots of the Characteristic Equation;
3.4 Repeated Roots; Reduction of Order
3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients;
3.6 Variation of Parameters
3.7 Mechanical and Electrical Vibrations
3.8 Forced Vibrations
Chapter 4 Higher Order Linear Equations;
4.1 General Theory of nth Order Linear Equations
4.2 Homogeneous Equations with Constant Coefficients;
4.3 The Method of Undetermined Coefficients;
4.4 The Method of Variation of Parameters
Chapter 5 Series Solutions of Second Order Linear Equations;
5.1;Review of Power Series;
5.2;Series Solutions Near an Ordinary Point, Part I;
5.3;Series Solutions Near an Ordinary Point, Part II;
5.4;Euler Equations; Regular Singular Points;
5.5;Series Solutions Near a Regular Singular Point, Part I;
5.6;Series Solutions Near a Regular Singular Point, Part II;
5.7; Bessel's Equation;
Chapter 6 The Laplace Transform
6.1;Definition of the Laplace Transform;
6.2;Solution of Initial Value Problems;
6.3;Step Functions;
6.4;Differential Equations with Discontinuous Forcing Functions;
6.5;Impulse Functions;
6.6;The Convolution Integral;
Chapter 7 Systems of First Order Linear Equations;
7.1;Introduction;
7.2;Review of Matrices;
7.3;Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors;
7.4;Basic Theory of Systems of First Order Linear Equations
7.5;Homogeneous Linear Systems with Constant Coefficients?;
7.6;Complex Eigenvalues;
7.7;Fundamental Matrices;
7.8;Repeated Eigenvalues;
7.9;Nonhomogeneous Linear Systems;
Chapter 8 Numerical Methods
8.1;The Euler or Tangent Line Method
8.2;Improvements on the Euler Method;
8.3;The Runge-Kutta Method
8.4;Multistep Methods;
8.5;More on Errors; Stability;
8.6;Systems of First Order Equations
Chapter 9 Nonlinear Differential Equations and Stability
9.1;The Phase Plane: Linear Systems;
9.2;Autonomous Systems and Stability;
9.3;Locally Linear Systems
9.4;Competing Species
9.5;Predator-Prey Equations;
9.6;Liapunov's Second Method;
9.7;Periodic Solutions and Limit Cycles;
9.8;Chaos and Strange Attractors: The Lorenz Equations;
Chapter10 Partial Differential Equations and Fourier Series
10.1 Two-Point Boundary Value Problems
10.2 Fourier Series;
10.3 The Fourier Convergence Theorem
10.4 Even and Odd Functions;
10.5 Separation of Variables; Heat Conduction in a Rod;
10.6 Other Heat Conduction Problems;
10.7 The Wave Equation: Vibrations of an Elastic String
10.8 Laplace's Equation;
Appendix A Derivation of the Heat Conduction Equation;
Appendix B Derivation of the Wave Equation;
Chapter 11 Boundary Value Problems and Sturm-Liouville Theory;
11.1 The Occurrence of Two-Point Boundary Value Problems;
11.2 Sturm-Liouville Boundary Value Problems;
11.3 Nonhomogeneous Boundary Value Problems;
11.4 Singular Sturm-Liouville Problems;
11.5 Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion;
11.6 Series of Orthogonal Functions: Mean Convergence;