Applied Differential Equations

Vladimir A. Dobrushkin is a Professor at the Division of Applied Mathematics, Brown University. He holds a Ph.D. in Applied mathematics and Dr.Sc. in mechanical engineering. He is the author of three books for CRC Press, including Applied Differential Equations with Boundary Value Problems, and Methods in Algorithmic Analysis. He is the co-author of Handbook of Differential Equations, Fourth Edition with Daniel Zwillinger.

As the author states in his preface, the Ordinary Differential Equations course is taught to undergraduates since some 200 years. If however a comparison is made between the book of Coddington and Levinson or the one of Hartman - two rather standard textbooks for at least half of century - and this one, the differences may be striking. The book aims to lay down a bridge between calculus,modeling and advanced topics (resulting from application requirements). Next the book emphasizes the role and the applications of qualitative theory. Among other aspects differentiating this book from others there can be mentioned: insertion of the topics from linear algebra as pre-requisite for the study of the systems of linear differential equations; discussion of the Sturm Liouville problems in two contexts: the orthogonal expansions and the boundary value problems occurring within the method of the separationof the variables for second order partial differential equations. Summarizing, this highly original textbook can introduce a new way of teaching ordinary differential equations. ~Vladimir Rasvan (Craiova) Sept. 2018

• Preface 1 Introduction1.1 Motivation 1.2 Classification of Differential Equations 1.3 Solutions to Differential Equations 1.4 Particular and Singular Solutions 1.5 Direction Fields 1.6 Existence and Uniqueness Review Questions for Chapter 1 2 First Order Equations 2.1 Separable Equations 2.1.1 Autonomous Equations 2.2 Equations Reducible to Separable Equations 2.2.1 Equations with Homogeneous Coefficients 2.2.2 Equations with Homogeneous Fractions 2.2.3 Equations with Linear Coefficients 2.3 Exact Differential Equations 2.4 Simple Integrating Factors 2.5 First-Order Linear Differential Equations 2.6 Special Classes of Equations 2.6.1 The Bernoulli Equation 2.6.2 The Riccati Equation 2.6.3 Equations with the Dependent or Independent Variable Missing2.6.4 Equations Homogeneous with Respect to Their Dependent Variable 2.6.5 Equations Solvable for a Variable 2.7 Qualitative Analysis 2.7.1 Bifurcation Points 2.7.2 Validity Intervals of Autonomous Equations Summary for Chapter 2 Review Questions for Chapter 2 3 Numerical Methods 3.1 Difference Equations 3.2 Euler’s Methods 3.3 The Polynomial Approximation 3.4 Error Estimates 3.5 The Runge–Kutta Methods Summary for Chapter 3 Review Questions for Chapter 3 4 Second and Higher Order Linear Differential Equations 4.1 Second and Higher Order Differential Equations 4.1.1 Linear Operators 4.1.2 Exact Equations and Integrating Factors 4.1.3 Change of Variables 4.2 Linear Independence and Wronskians 4.3 The Fundamental Set of Solutions 4.4 Equations with Constant Coefficients 4.5 Complex Roots 4.6 Repeated Roots. Reduction of Order 4.6.1 Reduction of Order 4.6.2 Euler’s Equations 4.7 Nonhomogeneous Equations 4.7.1 The Annihilator 4.7.2 The Method of Undetermined Coefficients 4.8 Variation of Parameters Summary for Chapter 4 Review Questions for Chapter 4 5 Laplace Transforms 5.1 The Laplace Transform 5.2 Properties of the Laplace Transform 5.3 Discontinuous and Impulse Functions 5.4 The Inverse Laplace Transform 5.4.1 Partial Fraction Decomposition 5.4.2 Convolution Theorem 5.4.3 The Residue Method 5.5 Homogeneous Differential Equations 5.5.1 Equations with Variable Coefficients 5.6 Nonhomogeneous Differential Equations 5.6.1 Differential Equations with Intermittent Forcing Functions Summary for Chapter 5 Review Questions for Chapter 5 6 Series Solutions of Differential Equations 3356.1 Power Series Solutions 6.2 Picard’s Iterations 6.3 Adomian Decomposition Method 6.4 Power Series Solutions to Equations with Analytic Coefficients 6.4.1 The Ordinary Point at Infinity 6.5 Euler Equations 6.6 Series Solutions Near a Regular Singular Point 6.6.1 Regular Singular Point at Infinity6.6.2 Inhomogeneous Equations 6.7 Bessel Equations 6.7.1 Parametric Bessel Equation 6.7.2 Bessel Functions of Half-Integer Order 6.7.3 Related Differential Equations Summary for Chapter 6Review Questions for Chapter 6 7 Introduction to Systems of ODEs 7.1 Some ODE Models 7.1.1 RLC-circuits 7.1.2 Spring-Mass Systems 7.1.3 The Euler–Lagrange Equation 7.1.4 Pendulum 7.1.5 Laminated Material 7.1.6 Flow Problems 7.2 Matrices 7.3 Linear Systems of First Order ODEs 7.4 Reduction to a Single ODE 7.5 Existence and Uniqueness Summary for Chapter 7 Review Questions for Chapter 7 8 Topics from Linear Algebra 8.1 The Calculus of Matrix Functions 8.2 Inverses and Determinants 8.2.1 Solving Linear Equations 8.3 Eigenvalues and Eigenvectors 8.4 Diagonalization 8.5 Sylvester’s Formula 8.6 The Resolvent Method 8.7 The Spectral Decomposition Method Summary for Chapter 8 Review Questions for Chapter 8 9 Systems of Linear Differential Equations 9.1 Systems of Linear Equations 9.1.1 The Euler Vector Equations 9.2 Constant Coefficient Homogeneous Systems 9.2.1 Simple Real Eigenvalues 9.2.2 Complex Eigenvalues 9.2.3 Repeated Eigenvalues 9.2.4 Qualitative Analysis of Linear Systems 9.3 Variation of Parameters 9.3.1 Equations with Constant Coefficients 9.4 Method of Undetermined Coefficients 9.5 The Laplace Transformation 9.6 Second Order Linear Systems Summary for Chapter 9 Review Questions for Chapter 9 10 Qualitative Theory of Differential Equations 10.1 Autonomous Systems 10.1.1 Two-Dimensional Autonomous Equations 10.2 Linearization 10.2.1 Two-Dimensional Autonomous Equations 10.2.2 Scalar Equations 10.3 Population Models10.3.1 Competing Species 10.3.2 Predator-Prey Equations 10.3.3 Other Population Models 10.4 Conservative Systems 10.4.1 Hamiltonian Systems 10.5 Lyapunov’s Second Method 10.6 Periodic Solutions 10.6.1 Equations with Periodic Coefficients Summary for Chapter 10 Review Questions for Chapter 10 11 Orthogonal Expansions11.1 Sturm–Liouville Problems 11.2 Orthogonal Expansions 11.3 Fourier Series 11.3.1 Music as Motivation 11.3.2 Sturm–Liouville Periodic Problem 11.3.3 Fourier Series 11.4 Convergence of Fourier Series 11.4.1 Complex Fourier Series 11.4.2 The Gibbs Phenomenon 11.5 Even and Odd Functions Summary for Chapter 11 Review Questions for Chapter 11 12 Partial Differential Equations 12.1 Separation of Variables for the Heat Equation 12.1.1 Two-Dimensional Heat Equation 12.2 Other Heat Conduction Problems 12.3 Wave Equation 12.3.1 Transverse Vibrations of Beams 12.4 Laplace Equation 12.4.1 Laplace Equation in Polar Coordinates Summary for Chapter 12 Review Questions for Chapter 12 Bibliography Index