Book:Morris Tenenbaum/Ordinary Differential Equations

Subject Matter

 * Ordinary Differential Equations

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



Historical Note on Radiocarbon Dating
Chapter $1$: Basic Concepts: Lesson $1$: How Differential Equations Originate

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

Source work progress
* : Chapter $1$: Basic Concepts: Lesson $2 \text C$: Function of Two Independent Variables: Example $2.62: 5$