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Calculus Ordinary Differential Equations by Morris Tenenbaum

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  1. Jan 19, 2013 #1

    Table of Contents:
    Code (Text):

    [LIST]
    [*] Preface for the Teacher


    [*] Preface for the Student


    [*] Basic Concepts

    [LIST]
    [*] How Differential Equations Originate.
    [*] The Meaning of the Terms Set and Function. Implicit Functions. Elementary Functions.
    [LIST]
    [*] The Meaning of the Term Set.
    [*] The Meaning of the Term Function of One Independent Variable.
    [*] Function of Two Independent Variables.
    [*] Implicit Function.
    [*] The Elementary Functions.
    [/LIST]
    [*] The Differential Equation.
    [LIST]
    [*] Definition of an Ordinary Differential Equation. Order of a Differential Equation.
    [*] Solution of a Differential Equation. Explicit Solution.
    [*] Implicit Solution of a Differential Equation.
    [/LIST]
    [*] The General Solution of a Differential Equation.
    [LIST]
    [*] Multiplicity of Solutions of a Differential Equation.
    [*] Method of Finding a Differential Equation if Its [itex]n[/itex]-Parameter Family of Solutions Is Known.
    [*] General Solution. Particular Solution. Initial Conditions.
    [/LIST]
    [*] Direction Field.
    [LIST]
    [*] Construction of a Direction Field. The Isoclines of a Direction Field.
    [*] The Ordinary and Singular Points of the First Order Equation (5.11).
    [/LIST]
    [/LIST]


    [*] Special Types of Differential Equations of the First Order
    [LIST]
    [*] Meaning of the Differential of a Function. Separable Differential Equations.
    [LIST]
    [*] Differential of a Function of One Independent Variable.
    [*] Differential of a Function of Two Independent Variables.
    [*] Differential Equations with Separable Variables.
    [/LIST]
    [*] First Order Differential Equation with Homogeneous Coefficients.
    [LIST]
    [*] Definition of a Homogeneous Function.
    [*] Solution of a Differential Equation in Which the Coefficients of [itex]dx[/itex] and [itex]dy[/itex] Are Each Homogeneous Functions of the Same Order.
    [/LIST]
    [*] Differential Equations with Linear Coefficients.
    [LIST]
    [*] Review of Some Plane Analytic Geometry.
    [*] Solution of a Differential Equation in Which the Coefficients of [itex]dx[/itex] and [itex]dy[/itex] are Linear, Nonhomogeneous, and When Equated to Zero Represent Non-parallel Lines.
    [*] A Second Method of Solving the Differential Equation (8.2) with Nonhomogeneous Coefficients.
    [*] Solution of a Differential Equation in Which the Coefficients of [itex]dx[/itex] and [itex]dy[/itex] Define Parallel or Coincident Lines.
    [/LIST]
    [*] Exact Differential Equations.
    [LIST]
    [*] Definition of an Exact Differential and of an Exact Differential Equation.
    [*] Necessary and Sufficient Condition for Exactness and Method of Solving an Exact Differential Equation.
    [/LIST]
    [*] Recognizable Exact Differential Equations. Integrating Factors.
    [LIST]
    [*] Recognizable Exact Differential Equations.
    [*] Integrating Factors.
    [*] Finding an Integrating Factor.
    [/LIST]
    [*] The Linear Differential Equation of the First Order. Bernoulli Equation.
    [LIST]
    [*] Definition of a Linear Differential Equation of the First Order.
    [*] Method of Solution of a Linear Differential Equation of the First Order.
    [*] Determination of the Integrating Factor [itex]e^{\int P(x)dx}[/itex].
    [*] Bernoulli Equation.
    [/LIST]
    [*] Miscellaneous Methods of Solving a First Order Differential Equation.
    [LIST]
    [*] Equations Permitting a Choice of Method.
    [*] Solution by Substitution and Other Means
    [/LIST]
    [/LIST]


    [*] Problems Leading to Differential Equations of the First Order
    [LIST]
    [*] Geometric Problems.
    [*] Trajectories.
    [LIST]
    [*] Isogonal Trajectories.
    [*] Orthogonal Trajectories.
    [*] Orthogonal Trajectory Formula in Polar Coordinates.
    [/LIST]
    [*] Dilution and Accretion Problems. Interest Problems. Temperature Problems. Decomposition and Growth Problems. Second Order Processes.
    [LIST]
    [*] Dilution and Accretion Problems.
    [*] Interest Problems.
    [*] Temperature Problems.
    [*] Decomposition and Growth Problems.
    [*] Second Order Processes.
    [/LIST]
    [*] Motion of a Particle Along a Straight Line — Vertical, Horizontal, Inclined.
    [LIST]
    [*] Vertical Motion.
    [*] Horizontal Motion.
    [*] Inclined Motion.
    [/LIST]
    [*] Pursuit Curves. Relative Pursuit Curves.
    [LIST]
    [*] Pursuit Curves.
    [*] Relative Pursuit Curve.
    [/LIST]
    [*] Miscellaneous Types of Problems Leading to Equations of the First Order
    [LIST]
    [*] Flow of Water Through an Orifice.
    [*] First Order Linear Electric Circuit.
    [*] Steady State Flow of Heat.
    [*] Pressure—Atmospheric and Oceanic.
    [*] Rope or Chain Around a Cylinder.
    [*] Motion of a Complex System.
    [*] Variable Mass. Rocket Motion.
    [*] Rotation of the Liquid in a Cylinder.
    [/LIST]
    [/LIST]


    [*] Linear Differential Equations of Order Greater Than One
    [LIST]
    [*] Complex Numbers and Complex Functions.
    [LIST]
    [*] Complex Numbers.
    [*] Algebra of Complex Numbers.
    [*] Exponential, Trigonometric, and Hyperbolic Functions of Complex Numbers.
    [/LIST]
    [*] Linear Independence of Functions. The Linear Differential Equation of Order [itex]n[/itex].
    [LIST]
    [*] Linear Independence of Functions.
    [*] The Linear Differential Equation of Order [itex]n[/itex]
    [/LIST]
    [*] Solution of the Homogeneous Linear Differential Equation of Order [itex]n[/itex] with Constant Coefficients.
    [LIST]
    [*] General Form of Its Solutions.
    [*] Roots of the Characteristic Equation (20.14) Real and Distinct.
    [*] Roots of Characteristic Equation (20.14) Real but Some Multiple.
    [*] Some or All Roots of the Characteristic Equation (20.14) Imaginary.
    [/LIST]
    [*] Solution of the Nonhomogeneous Linear Differential Equation of Order [itex]n[/itex] with Constant Coefficients.
    [LIST]
    [*] Solution by the Method of Undetermined Coefficients.
    [*] Solution by the Use of Complex Variables.
    [/LIST]
    [*] Solution of the Nonhomogeneous Linear Differential Equation by the Method of Variation of Parameters.
    [LIST]
    [*] Introductory Remarks.
    [*] The Method of Variation of Parameters.
    [/LIST]
    [*] Solution of the Linear Differential Equation with Nonconstant Coefficients. Reduction of Order Method.
    [LIST]
    [*] Introductory Remarks.
    [*] Solution of the Linear Differential Equation with Nonconstant Coefficients by the Reduction of Order Method.
    [/LIST]
    [/LIST]


    [*] Operators and Laplace Transforms
    [LIST]
    [*] Differential and Polynomial Operators.
    [LIST]
    [*] Definition of an Operator. Linear Property of Polynomial Operators.
    [*] Algebraic Properties of Polynomial Operators.
    [*] Exponential Shift Theorem for Polynomial Operators.
    [*] Solution of a Linear Differential Equation with Constant Coefficients by Means of Polynomial Operators.
    [/LIST]
    [*] Inverse Operators.
    [LIST]
    [*] Meaning of an Inverse Operator.
    [*] Solution of (25.1) by Means of Inverse Operators.
    [/LIST]
    [*] Solution of a Linear Differential Equation by Means of the Partial Fraction Expansion of Inverse Operators.
    [LIST]
    [*] Partial Fraction Expansion Theorem.
    [*] First Method of Solving a Linear Equation by Means of the Partial Fraction Expansion of Inverse Operators.
    [*] A Second Method of Solving a Linear Equation by Means of the Partial Fraction Expansion of Inverse Operators.
    [/LIST]
    [*] The Laplace Transform. Gamma Function.
    [LIST]
    [*] Improper Integral. Definition of a Laplace Transform.
    [*] Properties of the Laplace Transform.
    [*] Solution of a Linear Equation with Constant Coefficients by Means of a Laplace Transform.
    [*] Construction of a Table of Laplace Transforms.
    [*] The Gamma Function.
    [/LIST]
    [/LIST]


    [*] Problems Leading to Linear Differential Equations of Order Two
    [LIST]
    [*] Undamped Motion.
    [LIST]
    [*] Free Undamped Motion. (Simple Harmonic Motion.)
    [*] Definitions in Connection with Simple Harmonic Motion.
    [*] Examples of Particles Executing Simple Harmonic Motion. Harmonic Oscillators.
    [*] Forced Undamped Motion.
    [/LIST]
    [*] Damped Motion.
    [LIST]
    [*] Free Damped Motion. (Damped Harmonic Motion.)
    [*] Forced Motion with Damping.
    [/LIST]
    [*] Electric Circuits. Analog Computation.
    [LIST]
    [*] Simple Electric Circuit.
    [*] Analog Computation.
    [/LIST]
    [*] Miscellaneous Types of Problems Leading to Linear Equations of the Second Order
    [LIST]
    [*] Problems Involving a Centrifugal Force.
    [*] Rolling Bodies.
    [*] Twisting Bodies.
    [*] Bending of Beams.
    [/LIST]
    [/LIST]


    [*] Systems of Differential Equations. Linearization of First Order Systems.
    [LIST]
    [*] Solution of a System of Differential Equations.
    [LIST]
    [*] Meaning of a Solution of a System of Differential Equations.
    [*] Definition and Solution of a System of First Order Equations.
    [*] Definition and Solution of a System of Linear First Order Equations.
    [*] Solution of a System of Linear Equations with Constant Coefficients by the Use of Operators. Nondegenerate Case.
    [*] An Equivalent Triangular System.
    [*] Degenerate Case. [itex]f_1(D)g_2(D)-g_1(D)f_2(D)=0[/itex].
    [*] Systems of Three Linear Equations.
    [*] Solution of a System of Linear Differential Equations with Constant Coefficients by Means of Laplace Transforms.
    [/LIST]
    [*] Linearization of First Order Systems.
    [/LIST]


    [*] Problems Giving Rise to Systems of Equations. Special Types of Second Order Linear and Nonlinear Equations Solvable by Reducing to Systems.
    [LIST]
    [*] Mechanical, Biological, Electrical Problems Giving Rise to Systems of Equations.
    [LIST]
    [*] A Mechanical Problem — Coupled Springs.
    [*] A Biological Problem.
    [*] An Electrical Problem. More Complex Circuits.
    [/LIST]
    [*] Plane Motions Giving Rise to Systems of Equations.
    [LIST]
    [*] Derivation of Velocity and Acceleration Formulas.
    [*] The Plane Motion of a Projectile.
    [*] Definition of a Central Force. Properties of the Motion of a Particle Subject to a Central Force.
    [*] Definitions of Force Field, Potential, Conservative Field. Conservation of Energy in a Conservative Field.
    [*] Path of a Particle in Motion Subject to a Central Force Whose Magnitude Is Proportional
    to Its Distance from a Fixed Point O.
    [*] Path of a Particle in Motion Subject to a Central Force Whose Magnitude Is Inversely Proportional to the Square of Its Distance from a Fixed Point O.
    [*] Planetary Motion.
    [*] H. Kepler's (1571-1630) Laws of Planetary Motion. Proof of Newton's Inverse Square Law.
    [/LIST]
    [*] Special Types of Second Order Linear and Nonlinear Differential Equations Solvable by Reduction to a System of Two First Order Equations.
    [LIST]
    [*] Solution of a Second Order Nonlinear Differential Equation in Which [itex]y^\prime[/itex] and the Independent Variable [itex]x[/itex] Are Absent.
    [*] Solution of a Second Order Nonlinear Differential Equation in Which the Dependent Variable [itex]y[/itex] Is Absent.
    [*] Solution of a Second Order Nonlinear Equation in Which the Independent Variable [itex]x[/itex] Is Absent.
    [/LIST]
    [*] Problems Giving Rise to Special Types of Second Order Nonlinear Equations.
    [LIST]
    [*] The Suspension Cable.
    [*] A Special Central Force Problem.
    [*] A Pursuit Problem Leading to a Second Order Nonlinear Differential Equation.
    [*] Geometric Problems.
    [/LIST]
    [/LIST]


    [*] Series Methods
    [LIST]
    [*] Power Series Solutions of Linear Differential Equations.
    [LIST]
    [*] Review of Taylor Series and Related Matters.
    [*] Solution of Linear Differential Equations by Series Methods.
    [/LIST]
    [*] Series Solution of [itex]y^\prime = f(x,y)[/itex].
    [*] Series Solution of a Nonlinear Differential Equation of Order Greater Than One and of a System of First Order Differential Equations.
    [LIST]
    [*] Series Solution of a System of First Order Differential Equations.
    [*] Series Solution of a System of Linear First Order Equations.
    [*] Series Solution of a Nonlinear Differential Equation of Order Greater Than One.
    [/LIST]
    [*] Ordinary Points and Singularities of a Linear Differential Equation. Method of Frobenius.
    [LIST]
    [*] Ordinary Points and Singularities of a Linear Differential Equation.
    [*] Solution of a Homogeneous Linear Differential Equation About a Regular Singularity. Method of Frobenius.
    [/LIST]
    [*] The Legendre Differential Equation. Legendre Functions. Legendre Polynomials [itex]P_k(x)[/itex]. Properties of Legendre Polynomials [itex]P_k(x)[/itex].
    [LIST]
    [*] The Legendre Differential Equation.
    [*] Comments on the Solution (41.18) of the Legendre Equation (41.1). Legendre Functions.
    Legendre Polynomials [itex]P_k(x)[/itex].
    [*] Properties of Legendre Polynomials [itex]P_k(x)[/itex]
    [/LIST]
    [*] The Bessel Differential Equation. Bessel Function of the First Kind [itex]J_k(x)[/itex], Differential Equations Leading to a Bessel Equation. Properties of [itex]J_k(x)[/itex]
    [LIST]
    [*] The Bessel Differential Equation.
    [*] Bessel Functions of the First Kind [itex]J_k(x)[/itex].
    [*] Differential Equations Which Lead to a Bessel Equation.
    [*] Properties of Bessel Functions of the First Kind [itex]J_k(x)[/itex]
    [/LIST]
    [*] The Laguerre Differential Equation. Laguerre Polynomials [itex]L_k(x)[/itex]. Properties of [itex]L_k(x)[/itex]
    [LIST]
    [*] The Laguerre Differential Equation and Its Solution.
    [*] The Laguerre Polynomial [itex]L_k(x)[/itex].
    [*] Some Properties of Laguerre Polynomials [itex]L_k(x)[/itex]
    [/LIST]
    [/LIST]


    [*] Numerical Methods
    [LIST]
    [*] Starting Method. Polygonal Approximation.
    [*] An Improvement of the Polygonal Starting Method.
    [*] Starting Method — Taylor Series.
    [LIST]
    [*] Numerical Solution of [itex]y^\prime = f(x,y)[/itex] by Direct Substitution in a Taylor Series.
    [*] Numerical Solution of [itex]y^\prime = f(x,y)[/itex] by the "Creeping Up" Process.
    [/LIST]
    [*] Starting Method — Runge-Kutta Formulas.
    [*] Finite Differences. Interpolation.
    [LIST]
    [*] Finite Differences.
    [*] Polynomial Interpolation.
    [/LIST]
    [*] Newton's Interpolation Formulas.
    [LIST]
    [*] Newton's (Forward) Interpolation Formula.
    [*] Newton's (Backward) Interpolation Formula.
    [*] The Error in Polynomial Interpolation.
    [/LIST]
    [*] Approximation Formulas Including Simpson's and Weddle's Rule.
    [*] Milne's Method of Finding an Approximate Numerical Solution of [itex]y' = f(x,y)[/itex].
    [*] General Comments. Selecting [itex]h[/itex]. Reducing [itex]h[/itex]. Summary and an Example.
    [LIST]
    [*] Comment on Errors.
    [*] Choosing the Size of [itex]h[/itex].
    [*] Reducing and Increasing [itex]h[/itex].
    [*] Summary and an Illustrative Example.
    [/LIST]
    [*] Numerical Methods Applied to a System of Two First Order Equations.
    [*] Numerical Solution of a Second Order Differential Equation.
    [*] Perturbation Method. First Order Equation.
    [*] Perturbation Method. Second Order Equation.
    [/LIST]


    [*] Existence and Uniqueness Theorem for the First Order Differential Equation [itex]y^\prime= f(x,y)[/itex]. Picard's Method. Envelopes. Clairaut Equation.
    [LIST]
    [*] Picard's Method of Successive Approximations.
    [*] An Existence and Uniqueness Theorem for the First Order Differential Equation [itex]y^\prime = f(x,y)[/itex] Satisfying [itex]y(x_0)=y_0[/itex].
    [LIST]
    [*] Convergence and Uniform Convergence of a Sequence of Functions. Definition of a Continuous Function.
    [*] Lipschitz Condition. Theorems from Analysis.
    [*] Proof of the Existence and Uniqueness Theorem for the First Order Differential Equation [itex]y^\prime = f(x,y)[/itex].
    [/LIST]
    [*] The Ordinary and Singular Points of a First Order Differential Equation [itex]y^\prime = f(x,y)[/itex].
    [*] Envelopes.
    [LIST]
    [*] Envelopes of a Family of Curves.
    [*] Envelopes of a 1-Parameter Family of Solutions.
    [/LIST]
    [*] The Clairaut Equation.
    [/LIST]


    [*] Existence and Uniqueness Theorems for a System of First Order Differential equations and for Linear and Nonlinear Differential Equations of Order Greater Than One. Wronskians.
    [LIST]
    [*] An Existence and Uniqueness Theorem for a System of [itex]n[/itex] First Order Differential Equations and for a Nonlinear Differential Equation of Order Greater Than One.
    [LIST]
    [*] The Existence and Uniqueness Theorem for a System of [itex]n[/itex] First Order Differential Equations.
    [*] Existence and Uniqueness Theorem for a Nonlinear Differential Equation of Order [itex]n[/itex].
    [*] Existence and Uniqueness Theorem for a System of [itex]n[/itex] Linear First Order
    Equations.
    [/LIST]
    [*] Determinants. Wronskians.
    [LIST]
    [*] A Brief Introduction to the Theory of Determinants.
    [*] Wronskians.
    [/LIST]
    [*] Theorems About Wronskians and the Linear Independence of a Set of Solutions of a Homogeneous Linear Differential Equation.
    [*] Existence and Uniqueness Theorem for the Linear Differential Equation of Order [itex]n[/itex].
    [/LIST]


    [*] Bibliography


    [*] Index
    [/LIST]
     
     
    Last edited: May 6, 2017
  2. jcsd
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