# Book:Morris Tenenbaum/Ordinary Differential Equations

## Morris Tenenbaum and Harry Pollard: Ordinary Differential Equations

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

ISBN 0-486-64940-7.

### 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

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## 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,
 $\text {(a)}: \quad$ $\ds y$ $=$ $\ds 2,$ $\ds 0 \le x \le 1,$ $\ds y$ $=$ $\ds \sqrt x,$ $\ds 2 \le x \le 3.$
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 $2.211$