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A first integral for a class of time-dependent anharmonic oscillators with multiple anharmonicities. (English) Zbl 0761.34006

Summary: The integrability of the equation \(\ddot q=f(t)q^ 2+g(t)q^ 3+h(t)q^ 4+j(t)q^ 5\) is considered. Particular cases of this equation arise in the study of charged plasma in an axially symmetric magnetic field and in shear-free spherically symmetric gravitational fields in general relativity. The above equation with only a quadratic term arises in the study of shear-free fluids. The equation with a cubic term is applicable when there is an electromagnetic field. In special cases we reduce the solution to a quadrature that has solutions in terms of elliptic integrals. A Lie point symmetry analysis is performed and the different cases that arise are considered for the existence of a symmetry.

MSC:

34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
78A35 Motion of charged particles
34A05 Explicit solutions, first integrals of ordinary differential equations
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