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

Subject Matter

 * Ordinary Differential Equations

Contents

 * Preface (Cambridge, September 1955)
 * Preface to the Second Edition (May 1961)


 * $\text {I}$: EXISTENCE OF SOLUTIONS
 * 1. Some problems for investigation
 * 2. Simple ideas about solutions
 * 3. Existence of a solution
 * 4. Extensions of the existence theorem


 * $\text {II}$: 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


 * $\text {III}$: OSCILLATION THEOREMS
 * 11. Convexity of solutions
 * 12. Zeros of solutions
 * 13. Eigenvalues
 * 14. Eigenfunctions and expansions


 * $\text {IV}$: 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


 * $\text {V}$: 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


 * $\text {VI}$: 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 $\zeta - z$


 * $\text {VII}$: 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 $\nu$


 * $\text {VIII}$: 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


 * $\text {IX}$: 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 $\map {J_0} x$


 * $\text {I}$. The Laplace transform


 * $\text {II}$. Lines of force and equipotential surfaces









Source work progress
* : Chapter $\text I$: Existence of Solutions: $2$. Simple ideas about solutions: Example $1$.