The differential equation $Q=0$ in which $Q$ is a quadratic form in $yâ,yâ,y$ having meromorphic coefficients
HTML articles powered by AMS MathViewer
- by Roger Chalkley PDF
- Proc. Amer. Math. Soc. 116 (1992), 427-435 Request permission
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
-
P. Appell, Sur les équations différentielles algébriques et homogÚnes par rapport à la fonction inconnue et à ses dérivées, C. R. Acad. Sci. Paris 104 (1887), 1776-1779.
â, Sur une classe dâĂ©quations rĂ©ductibles aux Ă©quations linĂ©aires, C. R. Acad. Sci. Paris 107 (1888), 776-778.
â, Ăquations diffĂ©rentielles homogĂšnes du second ordre Ă coefficients constants, Ann. Fac. Sci. Toulouse Math. (1) 3 (1889), K1-K12.
â, Sur les invariants de quelques Ă©quations diffĂ©rentielles, J. Math. Pures Appl. (4) 5 (1889), 361-423.
- Ludwig Bieberbach, Theorie der gewöhnlichen Differentialgleichungen auf funktionentheoretischer Grundlage dargestellt, Die Grundlehren der mathematischen Wissenschaften, Band 66, Springer-Verlag, Berlin-New York, 1965 (German). Zweite umgearbeitete und erweiterte Auflage. MR 0176133
- Domenico Caligo, Sopra una classe di equazioni differenziali non lineari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 1 (1952), 24 pp. = Consiglio Naz. Ricerche. Pubbl. Ist. Appl. Calcolo no. 342 (1952) (Italian). MR 0052607
- Domenico Caligo, Sulla integrazione delle equazioni differenziali del secondo ordine a riferimento razionale, Univ. Roma Ist. Naz. Alta Mat. Rend. Mat. e Appl. (5) 11 (1952), (1953) (Italian). MR 54808
- Roger Chalkley, On the second order homogeneous quadratic differential equation, Math. Ann. 141 (1960), 87â98 (1960). MR 118870, DOI 10.1007/BF01367452
- Roger Chalkley, New contributions to the related work of Paul Appell, Lazarus Fuchs, Georg Hamel, and Paul PainlevĂ© on nonlinear differential equations whose solutions are free of movable branch points, J. Differential Equations 68 (1987), no. 1, 72â117. MR 885815, DOI 10.1016/0022-0396(87)90187-2
- Roger Chalkley, Relative invariants for homogeneous linear differential equations, J. Differential Equations 80 (1989), no. 1, 107â153. MR 1003253, DOI 10.1016/0022-0396(89)90098-3
- Christopher M. Cosgrove, New family of exact stationary axisymmetric gravitational fields generalising the Tomimatsu-Sato solutions, J. Phys. A 10 (1977), no. 9, 1481â1524. MR 503404, DOI 10.1088/0305-4470/10/9/010
- Christopher M. Cosgrove, A new formulation of the field equations for the stationary axisymmetric vacuum gravitational field. I. General theory, J. Phys. A 11 (1978), no. 12, 2389â2404. MR 513762, DOI 10.1088/0305-4470/11/12/007
- D. R. Curtiss, On the Invariants of a Homogeneous Quadratic Differential Equation of the Second Order, Amer. J. Math. 25 (1903), no. 4, 365â382. MR 1505924, DOI 10.2307/2370038
- J. J. Gergen and F. G. Dressel, Second order linear and nonlinear differential equations, Proc. Amer. Math. Soc. 16 (1965), 767â773. MR 180712, DOI 10.1090/S0002-9939-1965-0180712-7
- Robert Taylor Herbst, The equivalence of linear and nonlinear differential equations, Proc. Amer. Math. Soc. 7 (1956), 95â97. MR 76115, DOI 10.1090/S0002-9939-1956-0076115-0
- Murray S. Klamkin and James L. Reid, Nonlinear differential equations equivalent to solvable nonlinear equations, SIAM J. Math. Anal. 7 (1976), no. 3, 305â310. MR 399580, DOI 10.1137/0507024
- P. PainlevĂ©, Sur les Ă©quations diffĂ©rentielles du second ordre et dâordre supĂ©rieur dont lâintĂ©grale gĂ©nĂ©rale est uniforme, Acta Math. 25 (1902), no. 1, 1â85 (French). MR 1554937, DOI 10.1007/BF02419020
- Edmund Pinney, The nonlinear differential equation $y''+p(x)y+cy^{-3}=0$, Proc. Amer. Math. Soc. 1 (1950), 681. MR 37979, DOI 10.1090/S0002-9939-1950-0037979-4
- James 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), 61â62. MR 269907, DOI 10.1090/S0002-9939-1971-0269907-4
- James L. Reid, Homogeneous solution of a nonlinear differential equation, Proc. Amer. Math. Soc. 38 (1973), 532â536. MR 318542, DOI 10.1090/S0002-9939-1973-0318542-X
- J. M. Thomas, Equations equivalent to a linear differential equation, Proc. Amer. Math. Soc. 3 (1952), 899â903. MR 52001, DOI 10.1090/S0002-9939-1952-0052001-3
- P. R. Vein and P. Dale, Determinants, their derivatives and nonlinear differential equations, J. Math. Anal. Appl. 74 (1980), no. 2, 599â634. MR 572674, DOI 10.1016/0022-247X(80)90150-X G. Wallenberg, Ueber nichtlinear homogene Differentialgleichungen zweiter Ordnung, J. Reine Angew. Math. 119 (1898), 87-113.
Additional Information
- © Copyright 1992 American Mathematical Society
- 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