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First-Order Differential Equations | |
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Differential Equations and Mathematical Models | |
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Integrals as General and Particular Solutions | |
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Slope Fields and Solution Curves | |
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Separable Equations and Applications | |
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Linear First-Order Equations | |
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Substitution Methods and Exact Equations | |
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Mathematical Models and Numerical Methods | |
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Population Models | |
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Equilibrium Solutions and Stability | |
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Accelerationndash;Velocity Models | |
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Numerical Approximation: Euler's Method | |
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A Closer Look at the Euler Method | |
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The Rungendash;Kutta Method | |
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Linear Systems and Matrices | |
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Introduction to Linear Systems | |
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Matrices and Gaussian Elimination | |
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Reduced Row-Echelon Matrices | |
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Matrix Operations | |
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Inverses of Matrices | |
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Determinants | |
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Linear Equations and Curve Fitting | |
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Vector Spaces | |
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The Vector Space R3 | |
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The Vector Space Rn and Subspaces | |
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Linear Combinations and Independence of Vectors | |
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Bases and Dimension for Vector Spaces | |
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Row and Column Spaces | |
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Orthogonal Vectors in Rn | |
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General Vector Spaces | |
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Higher-Order Linear Differential Equations | |
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Introduction: Second-Order Linear Equations | |
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General Solutions of Linear Equations | |
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Homogeneous Equations with Constant Coefficients | |
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Mechanical Vibrations | |
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Nonhomogeneous Equations and Undetermined Coefficients | |
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Forced Oscillations and Resonance | |
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Eigenvalues and Eigenvectors | |
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Introduction to Eigenvalues | |
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Diagonalization of Matrices | |
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Applications Involving Powers of Matrices | |
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Linear Systems of Differential Equations | |
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First-Order Systems and Applications | |
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Matrices and Linear Systems | |
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The Eigenvalue Method for Linear Systems | |
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Second-Order Systems and Mechanical Applications | |
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Multiple Eigenvalue Solutions | |
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Numerical Methods for Systems | |
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Matrix Exponential Methods | |
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Matrix Exponentials and Linear Systems | |
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Nonhomogeneous Linear Systems | |
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Spectral Decomposition Methods | |
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Nonlinear Systems and Phenomena | |
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Stability and the Phase Plane | |
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Linear and Almost Linear Systems | |
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Ecological Models: Predators and Competitors | |
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Nonlinear Mechanical Systems | |
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Laplace Transform Methods | |
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Laplace Transforms and Inverse Transforms | |
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Transformation of Initial Value Problems | |
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Translation and Partial Fractions | |
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Derivatives, Integrals, and Products of Transforms | |
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Periodic and Piecewise Continuous Input Functions | |
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Power Series Methods | |
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Introduction and Review of Power Series | |
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Power Series Solutions | |
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Frobenius Series Solutions | |
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Existence and Uniqueness of Solutions | |
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Theory of Determinants Answers to Selected Problems | |
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Index | |
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Application Modules The modules listed here follow the indicated sections in the text | |
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Most provide computing projects that illustrate the corresponding text sections | |
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Maple, Mathematica, and MATLAB versions of these investigations are included in the Applications Manual that accompanies this textbook | |
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Computer-Generated Slope Fields and Solution Curves | |
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The Logistic Equation | |
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Indoor Temperature Oscillations | |
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Computer Algebra Solutions | |
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Logistic Modeling of Population Data | |
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Rocket Propulsion | |
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Implementing Euler's Method | |
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Improved Euler Implementation | |
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Runge-Kutta Implementation | |
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Automated Row Operations | |
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Automated Row Reduction | |
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Automated Solution of Linear Systems | |
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Plotting Seco | |