To the theory of geodesic lines equation. (Russian) Zbl 1058.53507

Lukashevich, N. A.; Chichurin, A. V.

Nelinijni Kolyvannya 2, No. 1, 30-35 (1999).

The authors determine a class of geodesic lines equations whose general integral is \[ \phi_3(x)=C_1\phi_1(x)\,\exp\lambda_1y(x) + C_2\phi_2(x)\,\exp\lambda_2y(x), \] where \(C_i\), \(\,i=1,2\), are arbitrary constants, \(\,\phi_i\), \(\,i=1,2,3\), are analytic functions \(\,(\not\equiv)\,\) and \(\,\lambda_1\) and \(\,\lambda_2\) are fixed constants. Also the Abel equation is considered. For this equation the conditions are established which ensure the integration in quadratures and the form of the general integral is shown.

Reviewer: A. Ju. Obolenskij (Kyïv)

MSC:

53C22	Geodesics in global differential geometry
34A30	Linear ordinary differential equations and systems, general

Keywords:

geodesic lines equation; general integral