Elementary Differential Equations
The Sixth Edition of this acclaimed differential equations book remains the same classic volume it's always been, but has been polished and sharpened to serve readers even more effectively. Offers precise and clear-cut statements of fundamental existence and uniqueness theorems to allow understanding of their role in this subject. Features a strong numerical approach that emphasizes that the effective and reliable use of numerical methods often requires preliminary analysis using standard...
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The Sixth Edition of this acclaimed differential equations book remains the same classic volume it's always been, but has been polished and sharpened to serve readers even more effectively. Offers precise and clear-cut statements of fundamental existence and uniqueness theorems to allow understanding of their role in this subject. Features a strong numerical approach that emphasizes that the effective and reliable use of numerical methods often requires preliminary analysis using standard elementary techniques. Inserts new graphics and text where needed for improved accessibility. A useful reference for readers who need to brush up on differential equations.
C O N T E N T SPreface viiCHAPTER1 First-Order Differential Equations 11.1 Differential Equations and Mathematical Models 11.2 Integrals as General and Particular Solutions 101.3 Slope Fields and Solution Curves 191.4 Separable Equations and Applications 321.5 Linear First-Order Equations 461.6 Substitution Methods and Exact Equations 591.7 Population Models 741.8 Acceleration-Velocity Models 85CHAPTER2 Linear Equations of Higher Order 1002.1 Introduction: Second-Order Linear Equations 1002.2 General Solutions of Linear Equations 1132.3 Homogeneous Equations with Constant Coefficients 1242.4 Mechanical Vibrations 1352.5 Nonhomogeneous Equations and Undetermined Coefficients 1482.6 Forced Oscillations and Resonance 1622.7 Electrical Circuits 1732.8 Endpoint Problems and Eigenvalues 180CHAPTER3 Power Series Methods 1943.1 Introduction and Review of Power Series 1943.2 Series Solutions Near Ordinary Points 2073.3 Regular Singular Points 2183.4 Method of Frobenius: The Exceptional Cases 2333.5 Bessel's Equation 2483.6 Applications of Bessel Functions 257vvi ContentsCHAPTER4 Laplace Transform Methods 2664.1 Laplace Transforms and Inverse Transforms 2664.2 Transformation of Initial Value Problems 2774.3 Translation and Partial Fractions 2894.4 Derivatives, Integrals, and Products of Transforms 2974.5 Periodic and Piecewise Continuous Input Functions 3044.6 Impulses and Delta Functions 316CHAPTER5 Linear Systems of Differential Equations 3265.1 First-Order Systems and Applications 3265.2 The Method of Elimination 3385.3 Matrices and Linear Systems 3475.4 The Eigenvalue Method for Homogeneous Systems 3665.5 Second-Order Systems and Mechanical Applications 3815.6 Multiple Eigenvalue Solutions 3935.7 Matrix Exponentials and Linear Systems 4075.8 Nonhomogeneous Linear Systems 420CHAPTER6 Numerical Methods 4306.1 Numerical Approximation: Euler's Method 4306.2 A Closer Look at the Euler Method 4426.3 The Runge-Kutta Method 4536.4 Numerical Methods for Systems 464CHAPTER7 Nonlinear Systems and Phenomena 4807.1 Equilibrium Solutions and Stability 4807.2 Stability and the Phase Plane 4887.3 Linear and Almost Linear Systems 5007.4 Ecological Models: Predators and Competitors 5137.5 Nonlinear Mechanical Systems 5267.6 Chaos in Dynamical Systems 542References for Further Study 555Appendix: Existence and Uniqueness of Solutions 559Answers to Selected Problems 573Index I-1
