Abstract:Although Jacobi elliptic functions have been known for almost two centuries, they are still the subject of intensive investigation. In this paper, contrary to the usual definition, we prove that the Jacobi elliptic functions can be defined by using nonconservative equations with limit cycles through existence theorems involving first integrals. This allows extending their validity domains, that is, their range of applications.
“…In this section, a brief review of the theory of second-order differential equations based on the first integral, recently introduced in the literature by the present authors, is presented [6,.…”
Generally, second-order differential equations are mapped onto first-order equations to determine their solutions. In the present paper, first-order differential equations with well-known analytic properties are transformed into new or known nonautonomous Lienard differential equations to obtain their exact solutions using the theory of second-order differential equations based on the existence of a first integral recently introduced in the literature by the present authors. First-order differential equations that appear in the Kamke book are used as illustrative examples. As a result, these examples show that parametrically excited Lienard equations may not exhibit parametric resonance.
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