ProofWiki:Books/E.L. Ince/Integration of Ordinary Differential Equations

E.L. Ince: Integration of Ordinary Differential Equations

Published 1939, 7th Editon 1956, Oliver and Boyd Ltd.

Subject Matter

• Ordinary Differential Equations

Contents

Preface (May 1939)

Preface to the Second Edition (April 1943) (A.C. Aitken and D.E. Rutherford)

CHAPTER I: EQUATIONS OF THE FIRST ORDER AND DEGREE

1. Definitions

2. Integration

3. Separation of Variables

4. The Homogeneous Type

5. The Equation with Linear Coefficients

6. Exact Equations

7. Integrating Factors

8. The Quotient of Two Integrating Factors

9. Special Types of Integrating Factor

10. The Linear Equation

11. The Bernoulli Equation

12. The Riccati Equation

13. Change of Variable

CHAPTER II: INTEGRAL CURVES

14. Families of Plane Curves

15. Trajectories

16. Level Lines and Lines of Slope on a Surface

17. Singular Points

CHAPTER III: EQUATIONS OF HIGHER DEGREE

18. The General Integral

19. The Clairaut Equation

20. Generalisation - the d'Alembert Equation

21. Further Generalisation

22. Equations with One Variable Missing

23. Homogeneous Equations

24. Geometrical Interpretation of a Differential Equation

25. Cusp on the Integral Curve

26. Envelope of Integral Curves

27. Equation of the Second Degree

CHAPTER IV: EQUATIONS OF THE SECOND AND HIGHER ORDERS

28. Reduction of the Order of an Equation

29. Equations that do not Involve $y$

30. Equations that do not Involve $x$

31. First Homogeneous Type

32. Second Homogeneous Type

33. Third Homogeneous Type

34. A Special Case of Homogeneity

35. First Integral

36. Problems Involving Curvature

CHAPTER V: LINEAR EQUATIONS

37. Form of the General Integral

38. Depression of the Order

39. Reduced Equation with Constant Coefficients

40. Properties of the Operator $F(D)$

41. Pairs of Conjugate Factors

42. Repeated Real Factors

43. Repeated Complex Factors

44. Inverse Operators

45. Inverse Operators Relative to a Periodic Function

46. Development of an Inverse Operator

47. General Integral by Quadratures

48. The Euler Linear Equation

49. The Laplace Linear Equation

50. Variation of Parameters

51. Linear Systems with Constant Coefficients

CHAPTER VI: SOLUTION IN SERIES

52. Solution Developed as a Taylor Series

53. Regular Singularity

54. The Hypergeometric Equation

55. The Legendre Equation and the Function $P_n \left({x}\right)$

56. Solution for Large Values of $\left|{x}\right|$

6%. The Bessel Equation and the Function $J_n \left({x}\right)$

58. The Function $Y_n \left({x}\right)$

EXAMPLES

SOLUTIONS

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