The modern landscape of technology and industry demands an equally modern approach to differential equations in the classroom. Designed for a first course in differential equations, the third edition of Brannan/Boyce’s Differential Equations: An Introduction to Modern Methods and Applications is consistent with the way engineers and scientists use mathematics in their daily work. The text emphasizes a systems approach to the subject and integrates the use of modern computing technology in the context of contemporary applications from engineering and science.

·First Order Differential Equations
·Systems of Two First Order Equations
·Second Order Linear Equations
·The Laplace Transform
·Systems of First Order Linear Equations
·Nonlinear Differential Equations and Stability
·Numerical Methods
·Series Solutions of Second order Equations
·Orthogonal Functions, Fourier Series and Boundary-Value Problems
·Elementary Partial Differential Equations


Table of Contents:
Chapter 1: Introduction 1.1 Mathematical Models and Solutions 1.2 Qualitative Methods: Phase Lines and Direction Fields 1.3 Definitions, Classification and Terminology Chapter 2: First Order Differential Equations 2.1 Separable Equations 2.2 Linear Equations: Method of Integrating Factors 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 Substitution Methods Chapter 3: Systems of Two First Order Equations 3.1 Systems of Two Linear Algebraic Equations 3.2 Systems of Two First Order Linear Differential Equations 3.3 Homogeneous Linear Systems with Constant Coefficients 3.4 Complex Eigenvalues 3.5 Repeated Eigenvalues 3.6 A Brief Introduction to Nonlinear Systems Chapter 4: Second Order Linear Equations 4.1 Definitions and Examples 4.2 Theory of Second Order Linear Homogeneous Equations 4.3 Linear Homogeneous Equations with Constant Coefficients 4.4 Mechanical and Electrical Vibrations 4.5 Nonhomogeneous Equations; Method of Undetermined Coefficients 4.6 Forced Vibrations, Frequency Response and Resonance 4.7 Variation of Parameters Chapter 5: The Laplace Transform 5.1 Definition of the Laplace Transform 5.2 Properties of the Laplace Transform 5.3 The Inverse Laplace Transform 5.4 Solving Differential Equations with Laplace Transforms 5.5 Discontinuous Functions and Periodic Functions 5.6 Differential Equations with Discontinuous Forcing Functions 5.7 Impulse Functions 5.8 Convolution Integrals and Their Applications 5.9 Linear Systems and Feedback Control Chapter 6: Systems of First Order Linear Equations 6.1 Definitions and Examples 6.2 Basic Theory of First Order Linear Systems 6.3 Homogeneous Linear Systems with Constant Coefficients 6.4 Non defective Matrices with Complex Eigenvalues 6.5 Fundamental Matrices and the Exponential of a Matrix 6.6 Nonhomogeneous Linear Systems 6.7 Defective Matrices Chapter 7: Nonlinear Differential Equations and Stability 7.1 Autonomous Systems and Stability 7.2 Almost Linear Systems 7.3 Competing Species 7.4 Predator-Prey Equations 7.5 Periodic Solutions and Limit Cycles 7.6 Chaos and Strange Attractors: The Lorenz Equations Chapter 8: Numerical Methods 8.1 Numerical Approximations: Euler's Method 8.2 Accuracy of Numerical Methods 8.3 Improved Euler and Runge-Kutta Methods 8.4 Numerical Methods for Systems of First Order Equations Chapter 9: Series Solutions of Second order Equations 9.1 Review of Power Series 9.2 Series Solutions Near an Ordinary Point, Part I 9.3 Series Solutions Near an Ordinary Point, Part II 9.4 Regular Singular Points 9.5 Series Solutions Near a Regular Singular Point, Part I 9.6 Series Solutions Near a Regular Singular Point, Part II 9.7 Bessel's Equation Chapter 10: Orthogonal Functions, Fourier Series and Boundary-Value Problems 10.1 Orthogonal Families in the Space PC [a,b] 10.2 Fourier Series 10.3 Elementary Two-Point Boundary Value Problems 10.4 General Sturm-Liouville Boundary Value Problems 10.5 Generalized Fourier Series and Eigenfunction Expansions 10.6 Singular Boundary Value Problems 10.7 Convergence Issues Chapter 11: Elementary Partial Differential Equations 11.1 Terminology 11.2 Heat Conduction in a Rod--Homogeneous Case 11.3 Heat Conduction in a Rod--Nonhomogeneous Case 11.4 Wave Equation--Vibrations of an Elastic String 11.5 Wave Equation--Vibrations of a Circular Membrane 11.6 Laplace Equation Appendices 11.A Derivation of the Heat Equation 11.B Derivation of the Wave Equation A: Matrices and Linear Algebra A.1 Matrices A.2 Systems of Linear Algebraic Equations, Linear Independence and Rank A.3 Determinates and Inverses A.4 The Eigenvalue Problem B: Complex Variables Answers to Selected Problems References Index