# Book:Morris Tenenbaum/Ordinary Differential Equations

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## Morris Tenenbaum and Harry Pollard:

## Contents

## Morris Tenenbaum and Harry Pollard: *Ordinary Differential Equations*

Published $\text {1963}$, **Dover Publications, Inc.**

- ISBN 0-486-64940-7.

### Subject Matter

### Contents

- PREFACE FOR THE TEACHER

- PREFACE FOR THE STUDENT

**1. BASIC CONCEPTS****Lesson 1. How Differential Equations Originate.**

**Lesson 2. The Meaning of the Terms***Set*and*Function*. Implicit Functions. Elementary Functions.**A.**The Meaning of the Term*Set*.**B.**The Meaning of the Term*Function of One Independent Variable*.**C.**Function of Two Independent Variables.**D.**Implicit Function.**E.**The Elementary Functions.

**Lesson 3. The Differential Equation.****A.**Definition of an Ordinary Differential Equation. Order of a Differential Equation.**B.**Solution of a Differential Equation. Explicit Solution.**C.**Implicit Solution of a Differential Equation.

**Lesson 4. The General Solution of a Differential Equation.****A.**Multiplicity of Solutions of a Differential Equation.**B.**Method of Finding a Differential Equation if Its $n$-parameter Family of Solutions Is Known.**C.**General Solution. Particular Solution. Initial Conditions.

**Lesson 5. Direction Field.****A.**Construction of a Direction Field. The Isoclines of a Direction Field.**B.**The Ordinary and Singular Points of the First Order Equation (5.11).

**2. SPECIAL TYPES OF DIFFERENTIAL EQUATIONS OF THE FIRST ORDER****Lesson 6. Meaning of the Differential of a Function. Separable Differential Equations.****A.**Differential of a Function of One Independent Variable.**B.**Differential of a Function of Two Independent Variables.**C.**Differential Equations with Separable Variables.

**Lesson 7. First Order Differential Equation with Homogeneous Coefficients.****A.**Definition of a Homogeneous Function.**B.**Solution of a Differential Equation in Which the Coefficients of $dx$ and $dy$ Are Each Homogeneous Functions of the Same Order.

**Lesson 8. Differential Equations with Linear Coefficients.****A.**A Review of Some Plane Analytic Geometry.**B.**Solution of a Differential Equation in Which the Coefficients of $dx$ and $dy$ are Linear, Nonhomogeneous, and When Equated to Zero Represent Nonparallel Lines.**C.**A Second Method of Solving the Differential Equation (8.2) with Nonhomogeneous Coefficients.**D.**Solution of a Differential Equation in Which the Coefficients of $dx$ and $dy$ Define Parallel or Coincident Lines.

**Lesson 9. Exact Differential Equations.****A.**Definition of an Exact Differential and of an Exact Differential Equation.**B.**Necessary and Sufficient Condition for Exactness and Method of Solving an Exact Differential Equation.

**Lesson 10. Recognizable Exact Differential Equations. Integrating Factors.****A.**Recognisable Exact Differential Equations.**B.**Integrating Factors.**C.**Finding an Integrating Factor.

**Lesson 11. The Linear Differential Equation of the First Order. Bernoulli Equation.****A.**Definition of a Linear Differential Equation of the First Order.**B.**Method of Solution of a Linear Differential Equation of the First Order.**C.**Determination of the Integrating Factor $e^{\int P \left({x}\right) dx}$.**D.**Bernoulli Equation.

**Lesson 12. Miscellaneous Methods of Solving a First Order Differential Equation.****A.**Equations Permitting a Choice of Method.**B.**Solution by Substitution and Other Means.

**3. PROBLEMS LEADING TO DIFFERENTIAL EQUATIONS OF THE FIRST ORDER****Lesson 13. Geometric problems.**

**Lesson 14. Trajectories.****A.**Isogonal Trajectories.**B.**Orthogonal Trajectories.**C.**Orthogonal Trajectory Formula in Polar Coordinates.

**Lesson 15. Dilution and Accretion Problems. Interest Problems. Temperature Problems. Decomposition and Growth Problems. Second Order Processes.****A.**Dilution and Accretion Problems.**B.**Interest Problems.**C.**Temperature Problems.**D.**Decomposition and Growth Problems.**E.**Second Order Processes.

**Lesson 16. Motion of a Particle Along a Straight Line - Vertical, Horizontal, Inclined.****A.**Vertical Motion.**B.**Horizontal Motion.**C.**Inclined Motion.

**Lesson 17. Pursuit Curves. Relative Pursuit Curves.****A.**Pursuit Curves.**B.**Relative Pursuit Curve.

**Lesson 17M. Miscellaneous Types of Problems Leading to Equations of the First Order****A.**Flow of Water Through an Orifice.**B.**First Order Linear Electric Circuit.**C.**Steady State Flow of Heat.**D.**Pressure - Atmospheric and Oceanic.**E.**Rope or Chain Around a Cylinder.**F.**Motion of a Complex System.**G.**Variable Mass. Rocket Motion.**H.**Rotation of the Liquid in a Cylinder.

**4. LINEAR DIFFERENTIAL EQUATIONS OF ORDER GREATER THAN ONE****Lesson 18. Complex Numbers and Complex Functions.****A.**Complex Numbers.**B.**Algebra of Complex Numbers.**C.**Exponential, Trigonometric, and Hyperbolic Functions of Complex Numbers.

**Lesson 19. Linear Independence of Functions. The Linear Differential Equation of Order $n$.****A.**Linear Independence of Functions.**B.**The Linear Differential Equation of Order $n$.

**Lesson 20. Solution of the Homogeneous Linear Differential Equation of Order $n$ with Constant Coefficients.****A.**General Form of Its Solutions.**B.**Roots of the Characteristic Equation (20.14) Real and Distinct.**C.**Roots of Characteristic Equation (20.14) Real but Some Multiple.**D.**Some or All Roots of the Characteristic Equation (20.14) Imaginary.

**Lesson 21. Solution of the Nonhomogeneous Linear Differential Equation of Order $n$ with Constant Coefficients.****A.**Solution by the Method of Undetermined Coefficients.**B.**Solution by the Use of Complex Variables.

**Lesson 22. Solution of the Nonhomogeneous Linear Differential Equation by the Method of Variation of Parameters.****A.**Introductory Remarks.**B.**The Method of Variation of Parameters.

**Lesson 23. Solution of the Linear Differential Equation with Nonconstant Coefficients. Reduction of Order Method.****A.**Introductory Remarks.**B.**Solution of the Linear Differential Equation with Nonconstant Coefficients by the Reduction of Order Method.

**5. OPERATORS AND LAPLACE TRANSFORMS****Lesson 24. Differential and Polynomial Operators.****A.**Definition of an Operator. Linear Property of Polynomial Operators.**B.**Algebraic Properties of Polynomial Operators.**C.**Exponential Shift Theorem for Polynomial Operators.**D.**Solution of a Linear Differential Equation with Constant Coefficients by Means of Polynomial Operators.

**Lesson 25. Inverse Operators.****A.**Meaning of an Inverse Operator.**B.**Solution of (25.1) by Means of Inverse Operators.

**Lesson 26. Solution of a Linear Differential Equation by Means of the Partial Fraction Expansion of Inverse Operators.****A.**Partial Fraction Expansion Theorem.**B.**First Method of Solving a Linear Equation by Means of the Partial Fraction Expansion of Inverse Operators.**C.**A Second Method of Solving a Linear Equation by Means of the Partial Fraction Expansion of Inverse Operators.

**Lesson 27. The Laplace Transform. Gamma Function.****A.**Improper Integral. Definition of a Laplace Transform.**B.**Properties of the Laplace Transform.**C.**Solution of a Linear Equation with Constant Coefficients by Means of a Laplace Transform.**D.**Construction of a Table of Laplace Transforms.**E.**The Gamma Function.

**6. PROBLEMS LEADING TO LINEAR DIFFERENTIAL EQUATIONS OF ORDER TWO****Lesson 28. Undamped Motion.****A.**Free Undamped Motion. (Simple Harmonic Motion.)**B.**Definitions in Connection with Simple Harmonic Motion.**C.**Examples of Particles Executing Simple Harmonic Motion. Harmonic Oscillators.**D.**Forced Undamped Motion.

**Lesson 29. Damped Motion.****A.**Free Damped Motion. (Damped Harmonic Motion.)**B.**Forced Motion with Damping.

**Lesson 30. Electric Circuits. Analog Computation.****A.**Simple Electric Circuit.**B.**Analog Computation.

**Lesson 30M. Miscellaneous Types of Problems Leading to Linear Equations of the Second Order****A.**Problems Involving a Centrifugal Force.**B.**Rolling Bodies.**C.**Twisting Bodies.**D.**Bending of Beams.

**7. SYSTEMS OF DIFFERENTIAL EQUATIONS. LINEARIZATION OF FIRST ORDER SYSTEMS****Lesson 31. Solution of a System of Differential Equations.****A.**Meaning of a Solution of a System of Differential Equations.**B.**Definition and Solution of a System of First Order Equations.**C.**Definition and Solution of a System of Linear First Order Equations.**D.**Solution of a System of Linear Equations with Constant Coefficients by the Use of Operators. Nondegenerate Case.**E.**An Equivalent Triangular System.**F.**Degenerate Case. $f_1 \left({D}\right) g_2 \left({D}\right) - g_1 \left({D}\right) f_2 \left({D}\right) = 0$.**G.**Systems of Three Linear Equations.**H.**Solution of a System of Linear Differential Equations with Constant Coefficients by Means of Laplace Transforms.

**Lesson 32. Linearization of First Order Systems.**

**8. PROBLEMS GIVING RISE TO SYSTEMS OF EQUATIONS. SPECIAL TYPES OF SECOND ORDER LINEAR AND NON-LINEAR EQUATIONS SOLVABLE BY REDUCING TO SYSTEMS****Lesson 33. Mechanical, Biological, Electrical Problems Giving Rise to Systems of Equations.****A.**A Mechanical Problem -- Coupled Springs.**B.**A Biological Problem.**C.**An Electrical Problem. More Complex Circuits.

**Lesson 34. Plane Motions Giving Rise to Systems of Equations.****A.**Derivation of Velocity and Acceleration Formulas.**B.**The Plane Motion of a Projectile.**C.**Definition of a Central Force. Properties of the Motion of a Particle Subject to a Central Force.**D.**Definitions of*Force Field*,*Potential*,*Conservative Field*. Conservation of Energy in a Conservative Field.**E.**Path of a Particle in Motion Subject to a Central Force Whose Magnitude Is Proportional to Its Distance from a Fixed Point $O$.**F.**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 $O$.**G.**Planetary Motion.**H.**Kepler's (1571-1630) Laws of Planetary Motion. Proof of Newton's Inverse Square Law.

**Lesson 35. Special Types of Second Order Linear and Nonlinear Differential Equations Solvable by Reduction to a System of Two First Order Equations.****A.**Solution of a Second Order Nonlinear Differential Equation in Which $y'$ and the Independent Variable $x$ Are Absent.**B.**Solution of a Second Order Nonlinear Differential Equation in Which the Dependent Variable $y$ Is Absent.**C.**Solution of a Second Order Nonlinear Equation in Which the Independent Variable $x$ Is Absent.

**Lesson 36. Problems Giving Rise to Special Types of Second Order Nonlinear Equations.****A.**The Suspension Cable.**B.**A Special Central Force Problem.**C.**A Pursuit Problem Leading to a Second Order Nonlinear Differential Equation.**D.**Geometric Problems.

**9. SERIES METHODS****Lesson 37. Power Serles Solutions of Linear Differential Equations.****A.**Review of Taylor Series and Related Matters.**B.**Solution of Linear Differential Equations by Series Methods.

**Lesson 38. Series Solution of $y' = f \left({x, y}\right)$.**

**Lesson 39. Series Solution of a Nonlinear Differential Equation of Order Greater Than One and of a System of First Order Differential Equations.****A.**Series Solution of a System of First Order Differential Equations.**B.**Series Solution of a System of Linear First Order Equations.**C.**Series Solution of a Nonlinear Differential Equation of Order Greater Than One.

**Lesson 40. Ordinary Points and Singularities of a Linear Differential Equation. Method of Frobenius.****A.**Ordinary Points and Singularities of a Linear Differential Equation.**B.**Solution of a Homogeneous Linear Differential Equation about a Regular Singularity. Method of Frobenius.

**Lesson 41. The Legendre Differential Equation. Legendre Functions. Legendre Polynomials $P_k \left({x}\right)$. Properties of Legendre Polynomials $P_k \left({x}\right)$.****A.**The Legendre Differential Equation.**B.**Comments on the Solution (41.18) of the Legendre Equation (41.1). Legendre Functions. Legendre Polynomials $P_k \left({x}\right)$.**C.**Properties of Legendre Polynomials $P_k \left({x}\right)$.

**Lesson 42. The Bessel Differential Equation. Bessel Function of the First Kind $J_k \left({x}\right)$. Differential Equations Leading to a Bessel Equation. Properties of $J_k \left({x}\right)$.****A.**The Bessel Differential Equation.**B.**Bessel Functions of the First Kind $J_k \left({x}\right)$.**C.**Differential Equations Which Lead to a Bessel Equation.**D.**Properties of Bessel Functions of the First Kind $J_k \left({x}\right)$.

**Lesson 43. The Laguerre Differential Equation. Laguerre Polynomials $L_k \left({x}\right)$. Properties of $L_k \left({x}\right)$.****A.**The Laguerre Differential Equation and Its Solution.**B.**The Laguerre Polynomial $L_k \left({x}\right)$.**C.**Some Properties of Laguerre Polynomials $L_k \left({x}\right)$.

**10. NUMERICAL METHODS****Lesson 44. Starting Method. Polygonal Approximation.**

**Lesson 45. An Improvement of the Polygonal Starting Method.**

**Lesson 46. Starting Method -- Taylor Series.****A.**Numerical Solution of $y' = f \left({x, y}\right)$ by Direct Substitution in a Taylor Series.**B.**Numerical Solution of $y' = f \left({x, y}\right)$ by the "Creeping Up" Process.

**Lesson 47. Starting Method-Runge-Kutta Formulas.**

**Lesson 48. Finite Differences. Interpolation.****A.**Finite Differences.**B.**Polynomial Interpolation.

**Lesson 49. Newton's Interpolation Formulas.****A.**Newton's (Forward) Interpolation Formula.**B.**Newton's (Backward) Interpolation Formula.**C.**The Error in Polynomial Interpolation.

**Lesson 50. Approximation Formulas Including Simpson's and Weddle's Rule.**

**Lesson 51. Milne's Method of Finding an Approximate Numerical Solution of $y' = f \left({x, y}\right)$.**

**Lesson 52. General Comments. Selecting $h$. Reducing $h$. Summary and an Example.****A.**Comment on Errors.**B.**Choosing the Size of $h$.**C.**Reducing and Increasing $h$.**D.**Summary and an Illustrative Example.

**Lesson 53. Numerical Methods Applied to a System of Two First Order Equations.**

**Lesson 54. Numerical Solution of a Second Order Differential Equation.**

**Lesson 55. Perturbation Method. First Order Equation.**

**Lesson 56. Perturbation Method. Second Order Equation.**

**11. EXISTENCE AND UNIQUENESS THEOREM FOR THE FIRST ORDER DIFFERENTIAL EQUATION $y' = f \left({x, y}\right)$. PICARD'S METHOD. ENVELOPES. CLAIRAUT EQUATION.****Lesson 57. Picard's Method of Successive Approximations.**

**Lesson 58. An Existence and Uniqueness Theorem for the First Order Differential Equation $y' = f \left({x, y}\right)$ Satisfying $y \left({x_0}\right) = y_0$.****A.**Convergence and Uniform Convergence of a Sequence of Functions. Definition of a Continuous Function.**B.**Lipschitz Condition. Theorems from Analysis.**C.**Proof of the Existence and Uniqueness Theorem for the First Order Differential Equation $y' = f \left({x, y}\right)$.

**Lesson 59. The Ordinary and Singular Points of a First Order Differential Equation $y' = f \left({x, y}\right)$.**

**Lesson 60. Envelopes.****A.**Envelopes of a Family of Curves.**B.**Envelopes of a 1-Parameter Family of Solutions.

**Lesson 61. The Clairaut Equation.**

**12. 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.****Lesson 62. 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.****A.**The Existence and Uniqueness Theorem for a System of $n$ First Order Differential Equations.**B.**Existence and Uniqueness Theorem for a Nonlinear Differential Equation of Order $n$.**C.**Existence and Uniqueness Theorem for a System of $n$ Linear First Order Equations.

**Lesson 63. Determinants. Wronskians.****A.**A Brief Introduction to the Theory of Determinants.**B.**Wronskians.

**Lesson 64. Theorems About Wronskians and the Linear Independence of a Set of Solutions of a Homogeneous Linear Differential Equation.**

**Lesson 65. Existence and Uniqueness Theorem for the Linear Differential Equation of Order $n$.**

**Bibliography**

**Index**

## Errata

### Historical Note on Radiocarbon Dating

Chapter $1$: Basic Concepts: Lesson $1$: How Differential Equations Originate

**Dr. Willard F. Libby was awarded the $1960$ Nobel Physics Prize for developing this method of ascertaining the age of ancient objects. His $C^{14}$ half-life figure is $5600$ years, ...*

### Arbitrary Function

Chapter $1$: Basic Concepts: Lesson $2 \text B$: The Meaning of the Term *Function of One Independent Variable*

*The relationship between two variables $x$ and $y$ is the following. If $x$ is between $0$ and $1$, $y$ is to equal $2$. If $x$ is between $2$ and $3$, $y$ is equal to $\sqrt x$. The equations which express the relationship between the two variables are, with the end points of the interval included,*

\(\text {(a)}: \quad\) | \(\displaystyle y\) | \(=\) | \(\displaystyle 2,\) | \(\displaystyle 0 \le x \le 1,\) | |||||||||

\(\displaystyle y\) | \(=\) | \(\displaystyle \sqrt x,\) | \(\displaystyle 2 \le x \le 3.\) |

*These*two*equations now define $y$ as a function of $x$. For each value of $x$ in the specified intervals, a value of $y$ is determined uniquely. The graph of this function is shown in Fig. $2.211$. Note that these equations do not define $y$ as a function of $x$ for values of $x$ outside the two stated intervals.*

**Figure $2.211$**

## Source work progress

- 1963: Morris Tenenbaum and Harry Pollard:
*Ordinary Differential Equations*... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $3$: The Differential Equation: Comment $3.53$