### Proof of regular local solvability of the Cauchy problem for differential equations in partial derivatives of the first order with initial data in Cartesian coordinates on line infinite length

#### L. E. Platonova^{1}

Annotation | The Cauchy problem for a quasi-linear first order partial differential equation is studied in case when initial data is given on an infinite length smooth line with non-vertical gradient. A system of integral equations, a solution of which gives a solution of the considered Cauchy problem in original coordinates and the solution has the same smoothness that the initial function, is constructed. Local solvability conditions, which do not include in itself assumptions about behavior of the characteristic lines, are presented in a theorem which proved here. |
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Keywords | quasi-linear first order partial differential equation, Cauchy problem, method of an additional argument |

^{1}The senior lecturer of the mathematics and mathematical education chair, Nizhniy Novgorod State Pedagogical University, Nizhniy Novgorod; fluff13@yandex.ru

**Citation**: L. E. Platonova, "[Proof of regular local solvability of the Cauchy problem for differential equations in partial derivatives of the first order with initial data in Cartesian coordinates on line infinite length]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,16:3 (2014) 77–86 (In Russian)