ProofWiki:Books/J.C. Burkill/The Theory of Ordinary Differential Equations

J.C. Burkill: The Theory of Ordinary Differential Equations

Published 1956, 2nd Editon 1962, Oliver and Boyd Ltd.

Subject Matter

• Ordinary Differential Equations

Contents

Preface (Cambridge, September 1955)

Preface to the Second Edition (May 1961)

CHAPTER I: EXISTENCE OF SOLUTIONS

1. Some problems for investigation

2. Simple ideas about solutions

3. Existence of a solution

4. Extensions of the existence theorem

CHAPTER II: THE LINEAR EQUATION

5. Existence theorem

6. The linear equation

7. Independent solutions

8. Solution of non-homogeneous equations

9. Second-order linear equations

10. Adjoint equations

CHAPTER III: OSCILLATION THEOREMS

11. Convexity of solutions

12. Zeros of solutions

13. Eigenvalues

14. Eigenfunctions and expansions

CHAPTER IV: SOLUTION IN SERIES

15. Differential equations in complex variables

16. Ordinary and singular points

17. Solutions near a regular singularity

18. Convergence of the power series

19. The second solution when exponents are equal or differ by an integer

20. The method of Frobenius

21. The point at infinity

22. Bessel's equation

CHAPTER V: SINGULARITIES OF EQUATIONS

23. Solutions near a singularity

24. Regular and irregular singularities

25. Equations with assigned singularities

26. The hypergeometric equation

27. The hypergeometric function

28. Expression of $F \left({a, b; c; z}\right)$ as an integral

29. Formulae connecting hypergeometric functions

30. Confluence of singularities

CHAPTER VI: CONTOUR INTEGRAL SOLUTIONS

31. Solutions expressed as integrals

32. Laplace's linear equation

33. Choice of contours

34. Further examples of contours

35. Integrals containing a power of $\zeta - z$

CHAPTER VII: LEGENDRE FUNCTIONS

36. Genesis of Legendre's equation

37. Legendre polynomials

38. Integrals for $P_n \left({z}\right)$

39. The generating function. Recurrence relations

40. The function $P_\nu \left({z}\right)$ for general $\nu$

CHAPTER VIII: BESSEL FUNCTIONS

41. Genesis of Bessel's equation

42. The solution $J_\nu \left({z}\right)$ in series

43. The generating function for $J_n \left({z}\right)$. Recurrence relations

44. Integrals for $J_\nu \left({z}\right)$

45. Contour integrals

46. Application of oscillation theorems

CHAPTER IX: ASYMPTOTIC SERIES

47. Asymptotic series

48. Definition and properties of asymptotic series

49. Asymptotic expansion of Bessel function

50. Asymptotic solutions of differential equations

51. Calculation of zeros of $J_0 \left({x}\right)$

APPENDIX I. The Laplace transform

APPENDIX II. Lines of force and equipotential surfaces

SOLUTIONS OF EXAMPLES

BIBLIOGRAPHY

INDEX