Tenenbaum, Morris & Pollard, Harry

Ordinary Differential Equations. An Elementary Textbook for Students of Mathematics, Engineering, and the Sciences

Harper & Row

[Harper's Mathematics Series]

New York 1963

#matematica

ig01#matematica

ig02#matematica

Privacy Policy

[i][c] INDICE:

0.15 | Preface for the teacher | ||

0.17 | Preface for the student | ||

1 | 1. | Basic concepts | |

1 | Lesson 1. | Ho Differential Equations Originate. | |

5 | Lesson 2. | The Meaning of the Terms Set and Function. Implicit Functions. Elementary Functions. | |

5 | A. | The Meaning of the Term Set. | |

6 | B. | The Meaning of the Term Function of One Independent Variable. | |

11 | C. | Function of Two Independent Variables. | |

14 | D. | Implicit Function. | |

17 | E. | The Elementary Functions | |

20 | Lesson 3. | The Differential Equation. | |

20 | A. | Definition of an Ordinary Differential Equation. Order of a Differential Equation. | |

21 | B. | Solution of a Differential Equation. Explicit Solution. | |

24 | C. | Implicit Solution of a Differential Equation | |

28 | Lesson 4. | The General Solution of a Differential Equation. | |

28 | A. | Multiplicity of Solutions of a Differential Equation. | |

31 | B. | Method of Finding a Differential Equation if Its n-Parameter Family of Solutions Is Known | |

33 | C. | General Solution. Particular Solution. Initial COnditions. | |

38 | Lesson 5. | Direction Field. | |

38 | A. | Construction of a Direction Field. The Isoclines of a Direction Field | |

41 | B. | The Ordinary and Singualr Points of the First Order Equation (5.11) | |

46 | 2. | Special types of differential equations of the first order | |

47 | Lesson 6. | Meaning of the Differential of a Function. Separable Differential Equations. | |

47 | A. | Differential of a Function of One Independent Variable | |

50 | B. | Differential of a Function of Two Independent Variables | |

51 | C. | Differential Equations with Separable Variables | |

57 | Lesson 7. | First Order Differential Equation with Homogeneous Coefficients. | |

57 | A. | Definition of a Homogeneous Function | |

58 | B. | Solution of a Differential Equation in Which the Coefficients of dx and dy Are Each Homogeneous Functions of the Same Order | |

62 | Lesson 8. | Differential Equations with Linear Coefficients | |

62 | A. | A Review of Some Plane Analytic Geometry. | |

63 | 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 | |

66 | C. | A Second Method of Solving the Differential Equation (8.2) with Nonhomogeneous Coefficients | |

67 | D. | Solution of a Differential Equation in Which the Coefficients of dx and dy Define Parallel or Coincident Lines. | |

70 | Lesson 9. | Exact Differential Equations. | |

72 | A. | Definition of an Exact Differential and of an Exact Differential Equation. | |

73 | B. | Necessary and Sufficient Condition for Exactness and Method of Solving an Exact Differential Equation. | |

80 | Lesson 10. | Recognizable Exact Differential Equations. Integrating Factors. | |

80 | A. | Recognizable Exact Differential Equations | |

82 | B. | Integrating Factors. | |

84 | C. | Finding an Integrating Factor. | |

91 | Lesson 11. | The Lineat Differential Equation of the First Order. Bernoulli Equation. | |

91 | A. | Definition of a Linear Differential Equation of the First Order | |

92 | B. | Method of Solution of a Linear Differential Equation of the First Order. | |

94 | C. | Determination of the Integrating Factor e^{Integrale[P(x)]dx} | |

95 | D. | Bernoulli Equation | |

99 | Lesson 12. | Miscellaneous Methods of Solving a First Order Differential Equation. | |

99 | A. | Equations Permitting a Choice of Method | |

101 | B. | Solution by Substitution and Other Means | |

107 | 3. | Problems leading to differential equations of the first order | |

107 | Lesson 13. | Geometric Problems. | |

115 | Lesson 14. | Trajectories. | |

115 | A. | Isogonal Trajectorie | |

117 | B. | Orthogonal Trajectories | |

118 | C. | Orthogonal Trajectory Formula in Polar Coordinates | |

122 | Lesson 15. | Dilution and Accretion Problems. Interest Problems. Temperature Problems. Decomposition and Growth Problems. Second Order Processes. | |

122 | A. | Diluition and Accretion Problems | |

126 | B. | Interest Problems | |

129 | C. | Temperature Problems | |

131 | D. | Decomposition and Growth Problems | |

134 | E. | Second Order Processes. | |

138 | Lesson 16. | Motion of a Particle Along a Straight Line - Vertical, Horizontal, Inclined. | |

139 | A. | Vertical Motion | |

160 | B. | Horizontal Motion. | |

164 | C. | Inclined Motion | |

168 | Lesson 17. | Pursuit Curves. Relative Pursuit Curves. | |

168 | A. | Pursuit Curves. | |

177 | B. | Relative Pursuit Curve | |

183 | Lesson 17M. | Miscellaneous Types of Problems Leading to Equations of the First Order. | |

183 | A. | Flow of Water Through an Orifice | |

184 | B. | First Order Linear Electric Circuit. | |

185 | C. | Steady State Flow of Heat. | |

186 | D. | Pressure - Atmospheric and Oceanic | |

188 | E. | Rope or Chian Around a Cylinder | |

189 | F. | Motion of a Complex System. | |

191 | G. | Variable Mass. Rocket Motion. | |

193 | H. | Rotation of the Liquid in a Cylinder | |

196 | 4. | Linear differential equations of order greater than one | |

197 | Lesson 18. | Complex Numbers and Complex Functions. | |

197 | A. | Complex Numbers. | |

200 | B. | Algebra of Complex Numbers | |

201 | C. | Exponential, Trigonometric, and Hyperbolic Functions of Complex Numbers | |

205 | Lesson 19. | Linear Independence of Functions. The Linear Differential Equation of Order n. | |

205 | A. | Linear Independence of Functions | |

207 | B. | The Linear Differential Equation of Order n | |

211 | Lesson 20. | Solution of the Homogeneous Linear Differential Equation of Order n with Constant Coefficients. | |

211 | A. | General Form and Its Solutions | |

213 | B. | Roots of the Characteristic Equation (20.14) Real and Distinct | |

214 | C. | Roots of Characteristic Equation (20.14) Real but Some Multiple. | |

217 | D. | Some or All Roots of the Characteristic Equation (20.14) Imaginary. | |

221 | Lesson 21. | Solution of the Nonhomogeneous Linear Differential Equation of Order n with Constant Coefficients. | |

221 | A. | Solution by the Method of Undeterminated Coefficients. | |

230 | B. | Solution by the Use of Complex Variables | |

233 | Lesson 22. | Solution of the Nonhomogeneous Linear Differential Equation by the Method of Variation of Parameters. | |

233 | A. | Introductory Remarks | |

233 | B. | The Method of Variation of Parameters. | |

241 | Lesson 23. | Solution of the Linear Differential Equation with Nonconstant Coefficients. Reduction of Order Method. | |

241 | A. | Introductory Remarks. | |

242 | B. | Solution of the Nonhomogeneous Linear Differential Equation with Nonconstant Coefficients by the Reduction of Order Method | |

250 | 5. | Operators and Laplace transforms | |

251 | Lesson 24. | Differential and Polynomial Operators. | |

251 | A. | Definition of an Operator. Linear Property of Polynomial Operators. | |

255 | B. | Algebraic Properties of Polynomial Operators | |

260 | C. | Exponential Shift Theorem for Polynomial Operators | |

262 | D. | Solution of a Linear Differential Equation with Constant Coefficients by Means of Polynomial Operators. | |

268 | Lesson 25. | Inverse Operators. | |

269 | A. | Meaning of an Inverse Operator. | |

272 | B. | Solution of (25.1) by Means of Inverse Operators. | |

283 | Lesson 26. | Solution of a Linear Differential Equation by Means of the Partial Fraction Expansion of Inverse Operators. | |

283 | A. | Partial Fraction Expansion Theorem | |

288 | B. | First Method of Solving a Linear Equation by Means of the Partial Fraction Expansion of Inverse Operators. | |

290 | C. | A Second Method of Solving a Linear Equation by Means of the Partial Fraction Expansion of Inverse Operators. | |

292 | Lesson 27. | The Laplace Transform. Gamma Function. | |

292 | A. | Improper Integral. Definition of a Laplace Transform. | |

295 | B. | Properties of the Laplace Transform. | |

296 | C. | Solution of a Linear Equation with Constant Coefficients by Means of a Laplace Transform | |

302 | D. | Construction of a Table of Laplace Transforms | |

306 | E. | The Gamma Function. | |

313 | 6. | Problems leading to linear differential equations of order two | |

313 | Lesson 28. | Undamped Motion. | |

313 | A. | Free Undamped Motion. | |

317 | B. | Definitions in Connection with Simple Harmonic Motion | |

323 | C. | Examples of Particles Executing Simple Harmonic Motion. Harmonic Oscillators | |

338 | D. | Forced Undamped Motion. | |

347 | Lesson 29. | Damped Motion. | |

347 | A. | Free Damped Motion. (Damped Harmonic Motion) | |

359 | B. | Forced Motion with Damping | |

369 | Lesson 30. | Electric Circuits. Analog Computation. | |

369 | A. | Simple Electric Circuit | |

375 | B. | Analog Computation | |

380 | Lesson 30M. | Miscellaneous Types of Problems Leading to Linear Equations of the Second Order. | |

380 | A. | Problems Involving a Centrifugal Force | |

381 | B. | Rolling Bodies | |

383 | C. | Twisting Bodies | |

383 | D. | Bending of Beams | |

393 | 7. | Systems of differential equations. Linearization of first order systems | |

393 | Lesson 31. | Solution of a System of Differential Equations. | |

393 | A. | Meaning of a Solution of a System of Differential Equations | |

394 | B. | Definition and Solution of a System of First Order Equations | |

396 | C. | Definition and Solution of a System of Linear First Order Equations | |

398 | D. | Solution of a System of Linear Equations with Constant COefficients by the Use of Operators. Nondegenerate Case. | |

405 | E. | An Equivalent Triangular System. | |

413 | F. | Degenerate Case. f_{1}(D) g_{2}(D) - g_{1}(D) f_{2}(D) = 0. | |

415 | G. | Systems of Three Linear Equations. | |

418 | H. | Solution of a System of Linear Differential Equations with Constant Coefficients by Means of Laplace Transforms | |

424 | Lesson 32. | Linearization of First Order Systems. | |

440 | 8. | Problems giving rise to systems of equations. Special types of second order linear and nonlinear equations solvable by reducing to systems | |

440 | Lesson 33. | Mechanical, Biological, Electrical Problems Giving Rise to Systems of Equations. | |

440 | A. | A Mechanical Problem - Coupled Springs | |

447 | B. | A Biological Problem. | |

451 | C. | An Electrical Problem. More Complex Circuits. | |

459 | Lesson 34. | Plane Motions Giving Rise to Systems of Equations. | |

459 | A. | Derivation of Velocity and Acceleration Formulas | |

463 | B. | The Plane Motion of a Projectile | |

470 | C. | Definition of a Central Force. Properties of the Motion of a Particle Subject to a Central Force. | |

473 | D. | Definitions of Force Field, Potential, Conservative Field. Conservation of Energy in a Conservative Field. | |

476 | E. | Path of a Particle in Motion Subject to a Central Force Whose Magnitude Is Proportional to Its Distance from a Fixed Point O. | |

481 | 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. | |

491 | G. | Planetary Motion. | |

492 | H. | Kepler's (1571-1630) Laws of Planetary Motion. Proof of Newton's Inverse Square Law. | |

500 | Lesson 35. | Special Types of Second Order Linear and Nonlinear Differential Equations Solvable by Reduction to a System of Two First Order Equations. | |

500 | A. | Solution of a Second Order Nonlinear Differential Equation in Which y' and the Independent Variable x Are Absent. | |

502 | B. | Solution of a Second Order Nonlinear Differential Equation in Which the Dependent Variable y Is Absent. | |

503 | C. | Solution of a Second Order Nonlinear Equation in Which the Independent Variable x is Absent. | |

506 | Lesson 36. | Problems Giving Rise to Special Types of Second Order Nonlinear Equations. | |

506 | A. | The Suspension Cable. | |

521 | B. | A Special Central Force Problem. | |

523 | C. | A Pursuit Problem Leading to a Second Order Nonlinear Differential Equation | |

528 | D. | Geometric Problems | |

531 | 9. | Series methods | |

531 | Lesson 37. | Power Series Solutions of Linear Differential Equations. | |

531 | A. | Review of Taylor Series and Related Matters. | |

537 | B. | Solution of Linear Differential Equations by Series Methods. | |

548 | Lesson 38. | Series Solution of y'=f(x,y). | |

555 | Lesson 39. | Series Solution of a Nonlinear Differential Equation of Order Greater Than One and of a System of First Order Differential Equations. | |

555 | A. | Series Solution of a System of First Order Differential Equations. | |

559 | B. | Series Solution of a System of Linear First Order Equations. | |

562 | C. | Series Solution of a Nonlinear Differential Equation of Order Greater Than One. | |

570 | Lesson 40. | Ordinary Points and Singularities of a Linear Differential Equation. Method of Frobenius. | |

570 | A. | Ordinary Points and Singularities of a Linear Differential Equation. | |

572 | B. | Solution of a Homogeneous Linear Differential Equation About a Regular Singularity. Method of Frobenius. | |

591 | Lesson 41. | The Legendre Differential Equation. Legendre Functions. Legendre Polynomials P_{k}(x). Properties of Legendre Polynomials P_{k}(x). | |

591 | A. | The Legendre Differential Equation. | |

593 | B. | Comments on the Solution (41.18) of the Legendre Equation (41.1). Legendre Functions. Legendre Polynomials P_{k}(x). | |

598 | C. | Properties of Legendre Polynomials P_{k}(x). | |

609 | Lesson 42. | 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). | |

609 | A. | The Bessel Differential Equation. | |

611 | B. | Bessel Functions of the First Kind J_{k}(x). | |

615 | C. | Differential Equations Which Lead to a Bessel Equation. | |

619 | D. | Properties of Bessel Functions of the First Kind J_{k}(x). | |

624 | Lesson 43. | The Laguerre Differential Equation. Laguerre Polynomials L_{k}(x). Properties of L_{k}(x) | |

624 | A. | The Laguerre Differential Equation and Its Solution. | |

625 | B. | The Laguerre Polynomial L_{k}(x). | |

627 | C. | Some Properties of Laguerre Polynomials L_{k}(x). | |

631 | 10. | Numerical methods | |

632 | Lesson 44. | Starting Method. Polygonal Approximation. | |

641 | Lesson 45. | An Improvement of the Polygonal Starting Method. | |

645 | Lesson 46. | Starting Method - Taylor Series. | |

646 | A. | Numerical Solution of y'=f(x,y) by Direct Substitution in a Taylor Series. | |

646 | B. | Numerical Solution of y'=f(x,y) by the "Creeping Up" Process. | |

653 | Lesson 47. | Starting Method - Runge-Kutta Formulas. | |

659 | Lesson 48. | Finite Differences. Interpolation. | |

659 | A. | Finite Differences. | |

661 | B. | Polynomial Interpolation. | |

663 | Lesson 49. | Newton's Interpolation Formulas. | |

663 | A. | Newton's (Forward) Interpolation Formula. | |

668 | B. | Newton's (Backward) Interpolation Formula. | |

670 | C. | The Error in Polynomial Interpolation. | |

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

684 | Lesson 51. | Milne's Method of FInding an Approximate Numerical Solution of y'=f(x,y) | |

690 | Lesson 52. | General Comments. Selecting h. Reducing h. Summary and an Example. | |

690 | A. | Comment on Errors | |

691 | B. | Choosing the Size of h. | |

692 | C. | Reducing and Increasing h. | |

694 | D. | Summary and an Illustrative Example. | |

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

707 | Lesson 54. | Numerical Solution of a Second Order Differential Equation. | |

713 | Lesson 55. | Pertubation Method. First Order Equation. | |

715 | Lesson 56. | Perturbation Method. second Order Equation. | |

719 | 11. | Existence and uniqueness theorem for the first order differential equation y'=f(x,y). Picard's method. Envelopes. Clairaut Equation. | |

720 | Lesson 57. | Picard's Method of Successive Approximaions. | |

728 | Lesson 58. | An Existence and Uniqueness Theorem for the FIrst Order Differential Equation y'=f(x,y) Satisfying y(x_{0}) = y_{0} | |

728 | A. | Convergence and Uniform Convergence of a Sequence of Functions. Definition of a Continuous Function. | |

731 | B. | Lipschitz Condition. Theorems from Analysis. | |

733 | C. | Proof of the Existence and Uniqueness Theorem for the First Order Differential Equation y'=f(x,y). | |

744 | Lesson 59. | The Ordinary and Singualr Points of a First Order Differential Equation y'=f(x,y). | |

747 | Lesson 60. | Envelopes. | |

748 | A. | Envelopes of a Family of Curves | |

754 | B. | Envelopes of a 1-Parameter Family of Solutions | |

757 | Lesson 61. | The Clairaut Equation. | |

763 | 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. | |

763 | 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. | |

763 | A. | The Existence and Uniqueness Theorem for a System of n First Order Differential Equations. | |

765 | B. | Existence and Uniqueness Theorem for a Nonlinear Differential Equation of Order n. | |

768 | C. | Existence and Uniqueness Theorem for a System of n Linear First Order Equations | |

770 | Lesson 63. | Determinants. Wronskians. | |

770 | A. | A Brief Introduction to the Theory of Determinants. | |

774 | B. | Wronskians. | |

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

783 | Lesson 65. | Existence and Uniquess Theorem for the Linear Differential Equation of Order n. | |

791 | Bibliography | ||

793 | Index |

[i][c] CRONOLOGIA:

Generato il giorno: 2020-08-10T17:27:52+02:00 (Unix Time: 1597073272)

Precedente aggiornamento il giorno: 0

Prima registrazione il giorno: 2020.0810

Aggiornato una volta

Dimensione approssimata della pagina: 68931 caratteri (body: 67267)

Versione: 1.0.46

Privacy Policy