# Construction of exact solutions to nonlinear equations of the hyperbolic type

 1Barannyk, AF, 2Barannyk, TA, 3Yuryk, II1Institute of Mathematics, Pomeranian University, Slupsk, Poland2V.G. Korolenko Poltava National Pedagogical University3National University of Food Technologies, Kyiv Dopov. Nac. akad. nauk Ukr. 2017, 7:3-9 https://doi.org/10.15407/dopovidi2017.07.003 Section: Mathematics Language: Ukrainian Abstract:  Substitutions that reduce the equation $u_{tt} = a(t)uu_{xx} + b(t)u_{x}^{2} + c(t)u$ to a system of ordinary differential equations are considered. An efficient method to integrate the corresponding reduced systems is proposed. It is shown that their integration can be reduced to the integration of a system of linear equations $w_{1}^{''} = \Phi_{1}(t)w_{1}$, $w_{2}^{''} = \Phi_{2}(t)w_{2}$, where $\Phi_{1}(t)$ and $\Phi_{2}(t)$ are arbitrary predefined functions. Keywords: differential equations, generalized separation of variables, nonlinear hyperbolic equations
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