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

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