Second-order differential equations: conditions of complete integrability. (English) Zbl 1199.34004
The original paper in Russian has been published in [Kiew. Nachr. 1880 (1880; JFM 12.0294.01)]. The paper was translated by A. O. Harin, under redaction by P. G. L. Leach.
In this remarkable paper, the integrability of the second-order equation \[ y''+Ay'+By=0\tag{1} \] is studied. It is known that, if we have a particular integral \(u=u(x)\) of (1), then one can find a general solution by quadratures. Ermakov gave conditions (with examples) under which equation (1) has a particular integral. The modern context of Ermakov’s results was presented by P. G. L. Leach and K. Andriopoulos in [Appl. Anal. Discrete Math. 2, No. 2, 146–157 (2008; Zbl 1199.34006)].
In this remarkable paper, the integrability of the second-order equation \[ y''+Ay'+By=0\tag{1} \] is studied. It is known that, if we have a particular integral \(u=u(x)\) of (1), then one can find a general solution by quadratures. Ermakov gave conditions (with examples) under which equation (1) has a particular integral. The modern context of Ermakov’s results was presented by P. G. L. Leach and K. Andriopoulos in [Appl. Anal. Discrete Math. 2, No. 2, 146–157 (2008; Zbl 1199.34006)].
Reviewer: Božidar Jovanović (Beograd)
MSC:
| 34A05 | Explicit solutions, first integrals of ordinary differential equations |
| 01A75 | Collected or selected works; reprintings or translations of classics |
