# Calculus Ordinary Differential Equations by Morris Tenenbaum

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1. Jan 19, 2013

### Greg Bernhardt

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 $n$-Parameter Family of Solutions Is Known.
[*] General Solution. Particular Solution. Initial Conditions.
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[*] 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]
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[*] 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 $dx$ and $dy$ 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 $dx$ and $dy$ 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 $dx$ and $dy$ 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 $e^{\int P(x)dx}$.
[*] 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 $n$.
[LIST]
[*] Linear Independence of Functions.
[*] The Linear Differential Equation of Order $n$
[/LIST]
[*] Solution of the Homogeneous Linear Differential Equation of Order $n$ 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 $n$ 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. $f_1(D)g_2(D)-g_1(D)f_2(D)=0$.
[*] 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 $y^\prime$ and the Independent Variable $x$ Are Absent.
[*] Solution of a Second Order Nonlinear Differential Equation in Which the Dependent Variable $y$ Is Absent.
[*] Solution of a Second Order Nonlinear Equation in Which the Independent Variable $x$ 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 $y^\prime = f(x,y)$.
[*] 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 $P_k(x)$. Properties of Legendre Polynomials $P_k(x)$.
[LIST]
[*] The Legendre Differential Equation.
[*] Comments on the Solution (41.18) of the Legendre Equation (41.1). Legendre Functions.
Legendre Polynomials $P_k(x)$.
[*] Properties of Legendre Polynomials $P_k(x)$
[/LIST]
[*] The Bessel Differential Equation. Bessel Function of the First Kind $J_k(x)$, Differential Equations Leading to a Bessel Equation. Properties of $J_k(x)$
[LIST]
[*] The Bessel Differential Equation.
[*] Bessel Functions of the First Kind $J_k(x)$.
[*] Differential Equations Which Lead to a Bessel Equation.
[*] Properties of Bessel Functions of the First Kind $J_k(x)$
[/LIST]
[*] The Laguerre Differential Equation. Laguerre Polynomials $L_k(x)$. Properties of $L_k(x)$
[LIST]
[*] The Laguerre Differential Equation and Its Solution.
[*] The Laguerre Polynomial $L_k(x)$.
[*] Some Properties of Laguerre Polynomials $L_k(x)$
[/LIST]
[/LIST]

[*] Numerical Methods
[LIST]
[*] Starting Method. Polygonal Approximation.
[*] An Improvement of the Polygonal Starting Method.
[*] Starting Method — Taylor Series.
[LIST]
[*] Numerical Solution of $y^\prime = f(x,y)$ by Direct Substitution in a Taylor Series.
[*] Numerical Solution of $y^\prime = f(x,y)$ 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 $y' = f(x,y)$.
[*] General Comments. Selecting $h$. Reducing $h$. Summary and an Example.
[LIST]
[*] Comment on Errors.
[*] Choosing the Size of $h$.
[*] Reducing and Increasing $h$.
[*] 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 $y^\prime= f(x,y)$. Picard's Method. Envelopes. Clairaut Equation.
[LIST]
[*] Picard's Method of Successive Approximations.
[*] An Existence and Uniqueness Theorem for the First Order Differential Equation $y^\prime = f(x,y)$ Satisfying $y(x_0)=y_0$.
[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 $y^\prime = f(x,y)$.
[/LIST]
[*] The Ordinary and Singular Points of a First Order Differential Equation $y^\prime = f(x,y)$.
[*] 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 $n$ 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 $n$ First Order Differential Equations.
[*] Existence and Uniqueness Theorem for a Nonlinear Differential Equation of Order $n$.
[*] Existence and Uniqueness Theorem for a System of $n$ 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 $n$.
[/LIST]

[*] Bibliography

[*] Index
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Last edited: May 6, 2017