Screen LibraryIntroduction to differential equations with dynamical system
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| Main Authors: | |
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| Group Author: | |
| Published: |
Princeton Univ. Press,
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| Publisher Address: | Princeton, N.J. |
| Publication Dates: | 2008. |
| Literature type: | Book |
| Language: | English |
| Subjects: | |
| Carrier Form: | xiii, 430 p.: ill. (some col.) ; 26 cm. |
| ISBN: |
0691124744 9780691124742 |
| Index Number: | O175 |
| CLC: |
O175 O193 |
| Call Number: | O175/C191 |
| Contents: |
Includes index. First-Order Differential Equations and Their Applications -- Introduction to ordinary differential equations -- The definite integral and the initial value problem -- The initial value problem and the indefinite integral -- The initial value problem and the definite integral -- Mechanics I: elementary motion of a particle with gravity only -- First-order separable differential equations -- Using definite integrals for separable differential equations -- Direction fields -- Existence and uniqueness -- Euler's numerical method (optional) -- First-order linear differential equations -- Form of the general solution -- Solutions of homogeneous first-order linear differential equations -- Integrating factors for first-order linear differential equations -- Linear first-order differential equations with constant coefficients and constant input -- Homogeneous linear differential equations with constant coefficients -- Constant coefficient linear differential equations with constant input -- Constant coefficient differential equations with exponential input -- Constant coefficient differential equations with discontinuous input -- Growth and decay rroblems -- A first model of population growth -- Radioactive decay -- Thermal cooling -- Mixture problems -- Mixture problems with a fixed volume -- Mixture problems with variable volumes -- Electronic circuits -- Mechanics II: including air Rrsistance -- Orthogonal trajectories (optional) -- Linear Second- and Higher-Order Differential Equations -- General solution of second-order linear differential equations -- Initial value problem (for homogeneous equations) -- Reduction of order -- Homogeneous linear constant coefficient differential equations (second order) -- Homogeneous linear constant coefficient differential equations (nth-order) -- Mechanical vibrations I: formulation and free response -- Formulation of equations -- Simple harmonic motion (no damping, 8 = 0) -- Free response with friction (8 > 0) -- The method of undetermined coefficients -- Mechanical vibrations II: forced response -- Friction is absent (8 = 0) -- Friction is present (8 > 0) (damped forced oscillations) -- Linear electric circuits -- Euler equation -- Variation of parameters (second-order) -- Variation of parameters (nth-order) -- The Laplace Transform -- Definition and basic properties -- The shifting theorem (multiplying by an exponential) -- Derivative theorem (multiplying by t) -- Inverse laplace transforms (roots, quadratics, and partial fractions) -- Initial value problems for differential equations -- Discontinuous forcing functions -- Solution of differential equations -- Periodic functions -- Integrals and the convolution theorem -- Derivation of the convolution theorem (optional) -- Impulses and distributions -- An Introduction to Linear Systems of Differential Equations and Their Phase Plane -- Introduction -- Introduction to linear systems of differential equations -- Solving linear systems using eigenvalues and eigenvectors of the matrix -- Solving linear systems if the eigenvalues are real and unequal -- Finding general solutions of linear systems in the case of complex eigenvalues -- Special systems with complex eigenvalues (optional) -- General solution of a linear system if the two real eigenvalues are equal (repeated) roots -- Eigenvalues and trace and determinant (optional) -- The phase plane for linear systems of differential equations -- Introduction to the phase plane for linear systems of differential equations -- Phase plane for linear systems of differential equations -- Real eigenvalues -- Complex eigenvalues -- General theorems -- Mostly Nonlinear First-Order Differential Equations -- First-order differential equations -- Equilibria and stability -- Equilibrium -- Stability -- Review of linearization -- Linear stability analysis -- One-dimensional phase lines -- Application to population dynamics: the logistic equation -- Nonlinear Systems of Differential Equations in the Plane -- Introduction -- Equilibria of nonlinear systems, linear stability analysis of equilibrium, and the phase plane -- Linear stability analysis and the phase plane -- Nonlinear systems: summary, philosophy, phase plane, direction field, nullclines -- Population models -- Two competing species -- Predator-prey population models -- Mechanical systems -- Nonlinear pendulum -- Linearized pendulum -- Conservative systems and the energy integral -- The phase plane and the potential -- Answers to odd-numbered exercises. |