# Book:J.C. Burkill/The Theory of Ordinary Differential Equations/Second Edition

## J.C. Burkill: The Theory of Ordinary Differential Equations (2nd Edition)

Published $\text {1962}$, Oliver and Boyd Ltd.

### Contents

Preface (Cambridge, September 1955)
Preface to the Second Edition (May 1961)
chapter $\text {I}$: EXISTENCE OF SOLUTIONS
1. Some problems for investigation
3. Existence of a solution
4. Extensions of the existence theorem
chapter $\text {II}$: THE LINEAR EQUATION
5. Existence theorem
6. The linear equation
7. Independent solutions
8. Solution of non-homogeneous equations
9. Second-order linear equations
chapter $\text {III}$: OSCILLATION THEOREMS
11. Convexity of solutions
12. Zeros of solutions
13. Eigenvalues
14. Eigenfunctions and expansions
chapter $\text {IV}$: 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 $\text {V}$: 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 $\map F {a, b; c; z}$ as an integral
29. Formulae connecting hypergeometric functions
30. Confluence of singularities
chapter $\text {VI}$: 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 $\text {VII}$: LEGENDRE FUNCTIONS
36. Genesis of Legendre's equation
37. Legendre polynomials
38. Integrals for $\map {P_n} z$
39. The generating function. Recurrence relations
40. The function $\map {P_\nu} z$ for general $\nu$
chapter $\text {VIII}$: BESSEL FUNCTIONS
41. Genesis of Bessel's equation
42. The solution $\map {J_\nu} z$ in series
43. The generating function for $\map {J_n} z$. Recurrence relations
44. Integrals for $\map {J_\nu} z$
45. Contour integrals
46. Application of oscillation theorems
chapter $\text {IX}$: 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 $\map {J_0} x$
Appendix $\text {I}$. The Laplace transform
Appendix $\text {II}$. Lines of force and equipotential surfaces
Solutions of Examples
Bibliography
Index

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