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

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