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Recent applications of the theory of Lie systems in Ermakov systems. (English) Zbl 1157.34029

The Milne-Pinney equation is investigated
\[ \ddot{x}=\omega^2(t)x+\frac{k}{x^3},\tag{1} \]
where \(k\) is a real constant with values depending on the field in which this equation is applied. Note, that Ermakov systems are systems of second order differential equations composed by the Milne-Pinney equation (1) together with the corresponding time-dependent harmonic oscillator. The authors review some recent results of the Lie systems theory to apply such results to study Ermakov systems. They show how superposition rule can be understood from a geometric point of view in some interesting cases, as the Milne-Pinney equation (1), the Ermakov system and its generalization. The known results about these differential equations have been recovered. Also new results have been obtained as a superposition rule allowing to write the general solution of the Pinney equation by means of three particular solutions of the Riccati equation.

MSC:

34C14 Symmetries, invariants of ordinary differential equations
34A26 Geometric methods in ordinary differential equations
34A05 Explicit solutions, first integrals of ordinary differential equations
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