| 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 eIntegrale[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. f1(D) g2(D) - g1(D) f2(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 Pk(x). Properties of Legendre Polynomials Pk(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 Pk(x). |
| 598 | C. | Properties of Legendre Polynomials Pk(x). |
| 609 | Lesson 42. | The Bessel Differential Equation. Bessel Function of the First Kind Jk(x). Differential Equations Leading to a Bessel Equation. Properties of Jk(x). |
| 609 | A. | The Bessel Differential Equation. |
| 611 | B. | Bessel Functions of the First Kind Jk(x). |
| 615 | C. | Differential Equations Which Lead to a Bessel Equation. |
| 619 | D. | Properties of Bessel Functions of the First Kind Jk(x). |
| 624 | Lesson 43. | The Laguerre Differential Equation. Laguerre Polynomials Lk(x). Properties of Lk(x) |
| 624 | A. | The Laguerre Differential Equation and Its Solution. |
| 625 | B. | The Laguerre Polynomial Lk(x). |
| 627 | C. | Some Properties of Laguerre Polynomials Lk(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(x0) = y0 |
| 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 |
