| Title:
|
Transformations of linear differential equations of second order and adjoined nonlinear equations (English) |
| Author:
|
Berkovich, L. M. |
| Author:
|
Rozov, N. H. |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
33 |
| Issue:
|
1 |
| Year:
|
1997 |
| Pages:
|
75-98 |
| . |
| Category:
|
math |
| . |
| MSC:
|
34-03 |
| MSC:
|
34A05 |
| MSC:
|
34A25 |
| MSC:
|
34C20 |
| idZBL:
|
Zbl 0914.34035 |
| idMR:
|
MR1464303 |
| . |
| Date available:
|
2008-06-06T21:32:36Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107599 |
| . |
| Reference:
|
[1] O. Borůvka: Lineare Differentialtransformationen 2.Ordnung.Berlin 1967. |
| Reference:
|
[2] O. Borůvka: Linear Differential Transformation of the Second Order.The English Universities Press, London 1971, 254 pages. MR 0463539 |
| Reference:
|
[3] O. Borůvka: Theory of global features of ordinary differential equations of second order.Differents. Uravneniya (in Russian) 12 (1976), 1347–1383. MR 0440123 |
| Reference:
|
[4] Otakar Borůvka: About oscillating integral of linear differential equations of second order.Czechoslovak Mathematical Journal, 3(78) (1953), N3, 199–255. |
| Reference:
|
[5] O. Borůvka: Grundlagen der Gruppoid und Gruppentheorie.Berlin 1960. MR 0117267 |
| Reference:
|
[6] V. I. Arnold: Supplementary chapters of theory of ordinary differential equations.Moscow 1978. MR 0526218 |
| Reference:
|
[7] L. M. Berkovich, N. H. Rozov: Some remarks about differential equations of form $y^{\prime \prime }+a_0(x)y=f(x)y^r$.Differents. Uravneniya (in Russian), 8 (1972), N11, 2076–2079. MR 0320407 |
| Reference:
|
[8] L. M. Berkovich: Transformation of ordinary differential equations.published at Kuibyshev University Press 1978, 92 p. MR 0548462 |
| Reference:
|
[9] L. M. Berkovich: About transformation of ordinary differential equations of Sturm-Liouville’s type.Functional Analysis and it’s Appl. (in Russian), 16 (1982), N3, 42–44. MR 0674007 |
| Reference:
|
[10] L. M. Berkovich: Factorization and transformation of ordinary differential equations.published at Saratov University Press, Saratov 1989, 192 p. MR 1063476 |
| Reference:
|
[11] L. M. Berkovich: Related linear differential equations of second order.Differents. Uravneniya (in Russian), 25 (1989), N2, 192–201. MR 0994698 |
| Reference:
|
[12] L. M. Berkovich: Canonical forms of ordinary linear differential equations.Arch. Math.(Brno), 24 (1988), N1, 25–42. Zbl 0672.34005, MR 0983005 |
| Reference:
|
[13] L. M. Berkovich: About one class of nonautonomial nonlinear differential equations of $n$-th order.Arch. Math. (Brno), 6 (1970), 7–13. |
| Reference:
|
[14] L. M. Berkovich, N. H. Rozov: Reduction to autonomous form some kind of nonlinear differential equations of second order.Arch. Math.(Brno) 8 (1972), 212–216. |
| Reference:
|
[15] L. M. Berkovich, N. H. Rozov: About Ermakov’s equation and some of its generalization.Modern Group Analysis and problems of mathematical modeling, XI Russian colloquium, Samara, 7–11, June 1993, Abstracts, Samara University Press, Samara 1993, p. 52. |
| Reference:
|
[16] L. M. Berkovich, N. H. Rozov: Ermakov’s equation: history and present time.Uspekhi Matem. Nauk (in Russian), 49 (1994), N4, p. 95. |
| Reference:
|
[17] G. D. Birkhoff: On the solutions of ordinary lineary homogeneous differential equations of the third order.Annals of Math., 12 (1910/11), 103–127. |
| Reference:
|
[18] P. Bohl: About some differential equations of general character.Applicable in mechanics (in Russian), Yuriev 1900, 114 p. (see also P. Bohl. Collection Works., Publisher “Zinatne”, Riega 1974, 73–198). |
| Reference:
|
[19] P. Bohl: Sur certaines equations differentielles d’un type general utilisables en mecanique.Bulletin de la Societe mathematique de France, 38 (1910), 1–134. |
| Reference:
|
[20] P. Bohl: Über ein Differentialgleichungen der Störungstheorie.Journal für die Reine und Angewandte Mathematik 131 (1906), H.4, 268–321. |
| Reference:
|
[] P. Bohl: Collection Works.Publisher “Zinatne”, Riega 1974, p. 327–377. |
| Reference:
|
[21] G. Darboux: Sur une proposition relative aux équations linéaires.C. R. Acad. Sci., Paris , 94 (1882), 1456–1459. |
| Reference:
|
[22] M. I. Elshin: On problem of oscillations second order linear differential equation.Doklady Acad. Nauk USSR, 18 (1938), N3, 141–145 (in Russian). |
| Reference:
|
[23] M. I. Elshin: Qualitative solution of second order linear differential equation.Uspekhi Matem. Nauk, 5 (1950), N2, 155–158 (in Russian). MR 0035892 |
| Reference:
|
[24] V. P. Ermakov: Second order differential equations.Integrability conditions in finite terms, Kiev, Universitetskiya Izvestiya 1880, N9, 1–25 (in Russian). |
| Reference:
|
[25] Euleri Leonardi: Methodus nova investigandi omnes casus quibus hanc aequationen differentio–differetialen $\partial d\partial y(1-axx)-bx\partial x\partial y-cy\partial x^2=0$.M.S.Academiae exhibit aie 13 Iannuarii 1780 (see also Institutiones calculi integralis, 4 (1794), 533–543). |
| Reference:
|
[26] A. R. Forsyth: Invariants, covariants and quotient-derivaties associated with linear differential equations.Philosophical Trans. of the Royal Society of London, 179A (1899) , 377–489. |
| Reference:
|
[27] I. M. Gelfand, L. A. Dikii: Sturm–Liouville’s equations asymptotics of resolvent and the algebra Korteveg–de Vries equations.Uspekhi Matem. Nauk, 30 (1975), N5(185), 67–100 (in Russian). MR 0508337 |
| Reference:
|
[28] M. Greguš: Lineárna diferenciálna rovnica tretieho rádu.Veda, Bratislava 1981. MR 0657356 |
| Reference:
|
[29] G.-H. Halphen: Mémoire sur la réduction des équations linéaires différentielles aux formes intégrables.Mémoires présentes par divers savants à l’Acad. des Sci., de l’inst. mat. de France, 28 (1884), N1, 1–301. |
| Reference:
|
[30] G. Hamel: Lineare Differentialgleichungen mit periodischen Koefficienten.Math. Ann. 73 (1913), 381. |
| Reference:
|
[31] F. Hanon: La transformation de Lyapunov de l’équation de Hill et son interpretation dynamique.Celestial Mechanics 28 (1982), 233–238. MR 0682852 |
| Reference:
|
[32] J. Heading: Transformations between second order linear differential equations....Proc. Roy. Soc., Edinburg A-79 (1977), N1–2, 87–105. Zbl 0372.34005, MR 0481193 |
| Reference:
|
[33] V. G. Imshenetskii: The extension in general linear equations of Euler’s method for research of all cases integrability of second order linear differential equations of special form.Zapiski Imperatorskoj Akademii Nauk, S.–Peterburg 42 (1882), 1–21 (in Russian). |
| Reference:
|
[34] E. E. Kummer: De generali quadam aequatione differentiali tertii ordinus.Abdruck aus dem Program des evangelischen Königl. und Stadtgymnasiums in Liegnitz von Jahre 1834; see also: J. Reine Angew. Math. 100 (1887), 1–9. |
| Reference:
|
[35] E. Laguerre: Sur les équations différentielles linéaires du troisième ordre.Comptes Rendus, Paris 88 1879, 116–118. |
| Reference:
|
[36] V. F. Lazutkin, T. V. Pankratova: The form normal and the deformation versals of the Hill’s equationFunctional Analysis and it’s Appl. 9 (1975), N4, 41–48 (in Russian).. MR 0467828 |
| Reference:
|
[37] J. Liouville: Sur le development des fonctions ou parties des fonctions en séries dont les divers termes sont se sujetti satisfaire à une meme équation différentielle du second ordre contenant un paramètre variable (second mémoire).J. Math. Pures et Appl. 2 (1837), 16–36. |
| Reference:
|
[38] A. M. Lyapunov: On question concerning of second order linear differential equations.Soobtsheniya Kharkov. matem. obtshestva, 2 ser., 1896–1897, N3–4, 5–6, 190–254 (in Russian), (see also A. M. Lyapunov, Collection Works, 2 (1956), 332–386 (in Russian)). |
| Reference:
|
[39] G. Mammana: Sopra un nuovo metodo di studio delle equazioni differenziali lineari.Math. Z. 25 (1926), 734–748. MR 1544837 |
| Reference:
|
[40] V. A. Marchenko: .Matem. Sbornik 95 (1974), N3, 331-356 (in Russian). Zbl 1167.01314 |
| Reference:
|
[41] F. Neuman: Categorial approach to global transformations of the $n$-th order linear differential equations.Časopis Pěst. Mat. 102 (1977), 350–355. Zbl 0374.34028, MR 0477284 |
| Reference:
|
[42] F. Neuman: Global properties of linear differential equations.Kluwer Acad. Publ.& Academia, Dordrecht–Berlin–London–Praha 1991. Zbl 0501.34003, MR 1192133 |
| Reference:
|
[43] E. Pinney: The Nonlinear Differential Equation $y^{\prime \prime }+p(t)y+cy^{-3}=0$.Proc. Amer. Math. Soc. 1 (1950), p. 581. MR 0037979 |
| Reference:
|
[44] A. V. Samohin: Symmetries Sturm–Liouville’s equations and Korteveg–de Vries equation.Doklady Akad. Nauk USSR 251 (1980), N3, 557–561 (in Russian). |
| Reference:
|
[45] S. Schneider, P. Winternitz: Classification of systems of nonlinear ordinary differential equations with superposition principles.J. Math. Phys. 25 (1984), N11, 3155–3165. MR 0761834 |
| Reference:
|
[46] P. Stäckel: Über Transformationen von Differentialgleichungen.J. Reine Angew. Math. 111 (1883), 290–302. |
| Reference:
|
[47] B. F. Whiting: The relation of solution of ODE’s is a commutation relative.Diff. Equat. Proc. Conf. Bratislava 1983. |
| Reference:
|
[48] V. A. Yakubovich: The questions of stability solutions of system two linear equations of canonical forms with periodic coefficients.Mat. Sb. 37 (1955), N1, 21–68 (in Russian). MR 0073769 |
| Reference:
|
[49] V. A. Yakubovich, V. M. Starzhinskii: Linear Differential Equations with Periodic Coefficients.Moscow, Nauka 1972 (in Russian). MR 0364739 |
| Reference:
|
[50] N. E. Joukovskii: The conditions of finiteness integrals of equation $d^2y/dx^2 + py=0$.Matem. Sb. 16 (1892), N3, 582–591 (in Russian). |
| . |
