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Full Description
For sophomore-level courses in Differential Equations and Linear Algebra.Extensively rewritten throughout, the Second Edition of this flexible text features a seamless integration of linear algebra into the discipline of differential equations. Abundant computer graphics, IDE interactive illustration software, and well-thought-out problem sets make it an excellent choice for either the combination DE/LA course or pure differential equations courses. The authors' consistent, reader-friendly presentation encourages students to think both quantitatively and qualitatively when approaching differential equations - and reinforces concepts using similar methods to solve various systems (algebraic, differential, and iterative).
Contents
1 First-Order Differential Equations1.1 Dynamical Systems: Modeling 11.2 Solutions and Direction Fields: Qualitative Analysis 111.3 Separation of Variables: Quantitative Analysis 251.4 Approximation Methods: Numerical Analysis 331.5 Picard's Theorem: Theoretical Analysis 462 Linearity and Nonlinearity2.1 Linear Equations: The Nature of Their Solutions 552.2 Solving the First-Order Linear Differential Equation 632.3 Growth and Decay Phenomena 732.4 Linear Models: Mixing and Cooling 802.5 Nonlinear Models: Logistic Equation 872.6 Systems of Differential Equations: A First Look 1003 Linear Algebra3.1 Matrices: Sums and Products 1153.2 Systems of Linear Equations 1303.3 The Inverse of a Matrix 1463.4 Determinants and Cramer's Rule 1563.5 Vector Spaces and Subspaces 1673.6 Basis and Dimension 1774 Higher-Order Linear Differential Equations4.1 The Harmonic Oscillator 1954.2 Real Characteristic Roots 2104.3 Complex Characteristic Roots 2294.4 Undetermined Coefficients 2444.5 Variation of Parameters 2554.6 Forced Oscillations 2614.7 Conservation and Conversion 2745 Linear Transformations5.1 Linear Transformations 2855.2 Properties of Linear Transformations 3005.3 Eigenvalues and Eigenvectors 3115.4 Coordinates and Diagonalization 3276 Linear Systems of Differential Equations6.1 Theory of Linear DE Systems 3436.2 Linear Systems with Real Eigenvalues 3576.3 Linear Systems with Nonreal Eigenvalues 3726.4 Stability and Linear Classification 3846.5 Decoupling a Linear DE System 3946.6 Matrix Exponential 4006.7 Nonhomogeneous Linear Systems 4107 Nonlinear Systems of Differential Equations7.1 Nonlinear Systems 4217.2 Linearization 4317.3 Numerical Solutions 4417.4 Chaos, Strange Attractors, and Period Doubling 4497.5 Chaos in Forced Nonlinear Systems 4568 Laplace Transforms8.1 The Laplace Transform and Its Inverse 4678.2 Solving DEs and IVPs with Laplace Transforms 4758.3 The Step Function and the Delta Function 4858.4 The Convolution Integral and the Transfer Function 4998.5 Laplace Transform Solution of Linear Systems 5099 Discrete Dynamical Systems9.1 Iterative Equations 5179.2 Linear Iterative Systems 5309.3 Nonlinear Iterative Equations: Chaos Again 54210 Control Theory10.1 Feedback Controls 55710.2 Introduction to Optimal Control 56710.3 Pontryagin Maximum Principle 579
