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Integrability of generalized Ermakov systems. (English) Zbl 0830.34002

Summary: We find all (four) first integrals for the two-dimensional Ermakov systems \[ \ddot x+ \omega^2(t) x= \textstyle{{1\over x^2}} f(y/x),\quad \ddot y+ \omega^2(t)y= \textstyle{{1\over y^3}} g(y/x). \] {}.

MSC:

34A05 Explicit solutions, first integrals of ordinary differential equations
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
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