Some exactly decided problems about non-linear oscillations

Doklady Bashkirskogo Universiteta. 2018. Volume 3. No. 2. pp. 162-166.

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Linearization is a widespread method in theory of differential equations. In particular, it is applied to description of oscillations with a non-linear retracting force. In this article, a procedure supplementing the linearization method is suggested. More exactly, in the Taylor expansion of the retracting force function the cubic term is taken into account. Then by selection of parameters, it is contraposed to a certain non-linear problem, retracting force of which coincides with the problem in question up to cubic terms. Solutions of the problem thus obtained are explicitly written through elementary functions.