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| First-Order Differential Equations | |
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| Differential Equations and Mathematical Model | |
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| Integrals as General and Particular Solutions | |
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| Direction 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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| Acceleration-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 Runge-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 R 3 | |
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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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| General Vector Spaces | |
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| Linear Equations of Higher Order | |
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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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| Undetermined Coefficients and Variation of Parameters | |
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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 Forcing 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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| Bessel's Equation | |
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| References for Further Study | |
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| The Existence and Uniqueness of Solutions | |
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| Theory of Determinants | |
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| Answers to Selected Problems | |
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| Index | |