Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The differential equation $ Q=0$ in which $ Q$ is a quadratic form in $ y'',y',y$ having meromorphic coefficients


Author: Roger Chalkley
Journal: Proc. Amer. Math. Soc. 116 (1992), 427-435
MSC: Primary 34A20; Secondary 34A05, 34C20
DOI: https://doi.org/10.1090/S0002-9939-1992-1112488-7
MathSciNet review: 1112488
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A simple necessary and sufficient condition is given for the solutions of $ Q = 0$ to be free of movable branch points. And, when the condition is satisfied, all the solutions of $ Q = 0$ can be obtained by solving linear differential equations of order $ \leq 2$. There are four mutually exclusive cases. We shall relate Case 4 to less convenient conditions P. Appell had introduced. We shall also show how Cases 3 and 4 together motivated our discovery of an identity that is essential for a satisfactory theory of relative invariants for homogeneous linear differential equations.


References [Enhancements On Off] (What's this?)

  • [1] P. Appell, Sur les équations différentielles algébriques et homogènes par rapport à la fonction inconnue et à ses dérivées, C. R. Acad. Sci. Paris 104 (1887), 1776-1779.
  • [2] -, Sur une classe d'équations réductibles aux équations linéaires, C. R. Acad. Sci. Paris 107 (1888), 776-778.
  • [3] -, Équations différentielles homogènes du second ordre à coefficients constants, Ann. Fac. Sci. Toulouse Math. (1) 3 (1889), K1-K12.
  • [4] -, Sur les invariants de quelques équations différentielles, J. Math. Pures Appl. (4) 5 (1889), 361-423.
  • [5] L. Bieberbach, Theorie der gewöhnlichen Differentialgleichungen auf funktionentheoretischer Grundlage dargestellt, 2nd ed., Springer-Verlag, Berlin, 1965. MR 0176133 (31:408)
  • [6] D. Caligo, Sopra una classe di equazioni differenziali non lineari, Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. (3) 1 (1952), 1-24. MR 0052607 (14:645f)
  • [7] -, Sulla integrazione delle equazioni differenziali del secondo ordine a riferimento razionale, Rend. Mat. Appl. (5) 11 (1952), 299-314. MR 0054808 (14:982e)
  • [8] R. Chalkley, On the second-order homogeneous quadratic differential equation, Math. Ann. 141 (1960), 87-98. MR 0118870 (22:9639)
  • [9] -, New contributions to the related work of Paul Appell, Lazarus Fuchs, Georg Hamel, and Paul Painlevé on nonlinear differential equations whose solutions are free of movable branch points, J. Differential Equations 68 (1987), 72-117. MR 885815 (88e:34010)
  • [10] -, Relative invariants for homogeneous linear differential equations, J. Differential Equations 80 (1989), 107-153. MR 1003253 (90e:34007)
  • [11] C. M. Cosgrove, New family of exact stationary axisymmetric gravitational fields generalizing the Tomimatsu-Sato solutions, J. Phys. A 10 (1977), 1481-1524. MR 0503404 (58:20168)
  • [12] -, A new formulation of the field equations for the stationary axisymmetric vacuum gravitational field I. General theory, J. Phys. A 11 (1978), 2389-2404. MR 513762 (80e:83036a)
  • [13] D. R. Curtiss, On the invariants of a homogeneous quadratic differential equation of the second order, Amer. J. Math. 25 (1903), 365-382. MR 1505924
  • [14] J. J. Gergen and F. G. Dressel, Second-order linear and nonlinear differential equations, Proc. Amer. Math. Soc. 16 (1965), 767-773. MR 0180712 (31:4943)
  • [15] R. T. Herbst, The equivalence of linear and nonlinear differential equations, Proc. Amer. Math. Soc. 7 (1956), 95-97. MR 0076115 (17:848c)
  • [16] M. S. Klamkin and J. L. Reid, Nonlinear differential equations equivalent to solvable nonlinear equations, SIAM J. Math. Anal. 7 (1976), 305-310. MR 0399580 (53:3423)
  • [17] P. Painlevé, Sur les équations différentielles du second ordre et d'ordre supérieur dont l'intégral générale est uniforme, Acta Math. 25 (1902), 1-86. MR 1554937
  • [18] E. Pinney, The nonlinear differential equation $ y'' + p(x)y + c{y^{ - 3}} = 0$, Proc. Amer. Math. Soc. 1 (1950), 681. MR 0037979 (12:336c)
  • [19] J. L. Reid, An exact solution of the nonlinear differential equation $ \ddot y + p(t)y = {q_m}(t)/{y^{2m - 1}}$, Proc. Amer. Math. Soc. 27 (1971), 61-62. MR 0269907 (42:4800)
  • [20] -, Homogeneous solution of a nonlinear differential equation, Proc. Amer. Math. Soc. 38 (1973), 532-536. MR 0318542 (47:7089)
  • [21] J. M. Thomas, Equations equivalent to a linear differential equation, Proc. Amer. Math. Soc. 3 (1952), 899-903. MR 0052001 (14:558d)
  • [22] P. R. Vein and P. Dale, Determinants, their derivatives and nonlinear differential equations, J. Math. Anal. Appl. 74 (1980), 599-634. MR 572674 (81f:15013)
  • [23] G. Wallenberg, Ueber nichtlinear homogene Differentialgleichungen zweiter Ordnung, J. Reine Angew. Math. 119 (1898), 87-113.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 34A20, 34A05, 34C20

Retrieve articles in all journals with MSC: 34A20, 34A05, 34C20


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1112488-7
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society