Book:J.C. Burkill/The Theory of Ordinary Differential Equations/Second Edition
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J.C. Burkill: The Theory of Ordinary Differential Equations (2nd Edition)
Published $\text {1962}$, Oliver and Boyd Ltd
Subject Matter
Contents
- Preface (Cambridge, September 1955)
- Preface to the Second Edition (May 1961)
- chapter $\text {I}$: 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 $\text {II}$: THE LINEAR EQUATION
- 5. Existence theorem
- 6. The linear equation
- 7. Independent solutions
- 8. Solution of non-homogeneous equations
- 9. Second-order linear equations
- 10. Adjoint 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
Further Editions
Source work progress
- 1962: J.C. Burkill: The Theory of Ordinary Differential Equations (2nd ed.) ... (previous) ... (next): Chapter $\text I$: Existence of Solutions: $2$. Simple ideas about solutions: Example $1$.