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| Introduction | |
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| Essentials of linear algebra | |
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| Motivating problems | |
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| Systems of linear equations | |
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| Row reduction using Maple | |
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| Linear combinations | |
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| Markov chains: an application of matrix-vector multiplication | |
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| Matrix products using Maple | |
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| The span of a set of vectors | |
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| Systems of linear equations revisited | |
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| Linear independence | |
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| Matrix algebra | |
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| Matrix algebra using Maple | |
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| The inverse of a matrix | |
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| Computer graphics | |
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| Matrix inverses using Maple | |
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| The determinant of a matrix | |
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| Determinants using Maple | |
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| The eigenvalue problem | |
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| Markov chains, eigenvectors, and Google | |
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| Using Maple to find eigenvalues and eigenvectors | |
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| Generalized vectors | |
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| Bases and dimension in vector spaces | |
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| For further study | |
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| Computer graphics: geometry and linear algebra at work | |
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| B�zier curves | |
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| Discrete dynamical systems | |
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| First-order differential equations | |
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| Motivating problems | |
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| Definitions, notation, and terminology | |
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| Plotting slope fields using Maple | |
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| Linear first-order differential equations | |
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| Applications of linear first-order differential equations | |
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| Mixing problems | |
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| Exponential growth and decay | |
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| Newton's law of Cooling | |
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| Nonlinear first-order differential equations | |
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| Separable equations | |
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| Exact equations | |
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| Euler's method | |
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| Implementing Euler's method in Excel | |
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| Applications of nonlinear first-order differential equations | |
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| The logistic equation | |
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| Torricelli's law | |
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| For further study | |
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| Converting certain second-order des to first-order DEs | |
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| How raindrops fall | |
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| Riccati's equation | |
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| Bernoulli's equation | |
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| Linear systems of differential equations | |
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| Motivating problems | |
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| The eigenvalue problem revisited | |
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| Homogeneous linear first-order systems | |
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| Systems with all real linearly independent eigenvectors | |
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| Plotting direction fields for systems using Maple | |
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| When a matrix lacks two real linearly independent eigenvectors | |
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| Nonhomogeneous systems: undetermined coefficients | |
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| Nonhomogeneous systems: variation of parameters | |
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| Applying variation of parameters using Maple | |
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| Applications of linear systems | |
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| Mixing problems | |
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| Spring-mass systems | |
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| RLC circuits | |
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| For further study | |
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| Diagonalizable matrices and coupled systems | |
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| Matrix exponential | |
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| Higher order differential equations | |
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| Motivating equations | |
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| Homogeneous equations: distinct real roots | |
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| Homogeneous equations: repeated and complex roots | |
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| Repeated roots | |
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| Complex roots | |
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| Nonhomogeneous equations | |
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| Undetermined coefficients | |
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| Variation of parameters | |
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| Forced motion: beats and resonance | |
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| Higher order linear differential equations | |
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| Solving characteristic equations using Maple | |
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| For further study | |
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| Damped motion | |
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| Forced oscillations with damping | |
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| The Cauchy-Euler equation | |
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| Companion systems and companion matrices | |
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| Laplace transforms | |
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| Motivating problems | |
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| Laplace transforms: getting started | |
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| General properties of the Laplace transform | |
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| Piecewise continuous functions | |
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| The Heaviside functions | |
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| The Dirac delta function | |
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| The Heaviside and Dirac functions in Maple | |
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| Solving IVPs with the Laplace transform | |
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| More on the inverse Laplace transform | |
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| Laplace transforms and inverse transforms using Maple | |
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| For further study | |
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| Laplace transforms of infinite series | |
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| Laplace transforms of periodic forcing functions | |
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| Laplace transforms of systems | |
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| Nonlinear systems of differential equations | |
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| Motivating problems | |
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| Graphical behavior of solutions for 2 � 2 nonlinear systems | |
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| Plotting direction fields of nonlinear systems using Maple | |
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| Linear approximations of nonlinear systems | |
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| Euler's method for nonlinear systems | |
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| Implementing Euler's method for systems in Excel | |
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| For further study | |
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| The damped pendulum | |
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| Competitive species | |
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| Numerical methods for differential equations | |
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| Motivating problems | |
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| Beyond Euler's method | |
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| Heun's method | |
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| Modified Euler's method | |
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| Higher order methods | |
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| Taylor methods | |
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| Runge-Kutta methods | |
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| Methods for systems and higher order equations | |
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| Euler's methods for systems | |
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| Heun's method for systems | |
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| Runge-Kutta method for systems | |
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| Methods for higher order IVPs | |
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| For further study | |
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| Predator-Prey equations | |
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| Competitive species | |
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| The damped pendulum | |
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| Series solutions for differential equations | |
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| Motivating problems | |
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| A review of Taylor and power series | |
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| Power series solutions of linear equations | |
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| Legendre's equation | |
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| Three important examples | |
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| The Hermite equation | |
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| The Laguerre equation | |
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| The Bessel equation | |
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| The method of Frobenius | |
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| For further study | |
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| Taylor series for first-order differential equations | |
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| The Gamma function | |
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| Review of integration techniques | |
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| Complex numbers | |
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| Roots of polynomials | |
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| Linear transformations | |
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| Solutions to selected exercises | |
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| Index | |