Two-dimensional non-autonomous hyperbolic equation with quadratic polynomial on first derivatives

There is investigated two-dimensional non-autonomous hyperbolic equation, the right side of which contains arbitrary non-linearity on unknown function and the quadratic polynomial on its first derivatives. The solutions of this equation are received in explicit form for the simplest nonlinearities with the help of the methods of multiplicative and functional separation of variables. It is showed that under certain conditions the initial equation can be reduced to the quadratic equation with respect to some auxiliary variable. There is received the solution as a quadratic form on some functions of one variable, and also the solution in the form of the production of powers on independent variables for the case when the coefficients of initial equation are the power functions. It is showed by means of Clarkson - Kruskal method, that the initial equation can be reduced to the Riccati equation with constant coefficients in the case when the coefficients of initial equation are expressed through the relation of functions of one variable. The corresponding exact solutions in explicit form are received. There is received the exact solution in implicit form for the case of arbitrary non-linearity on unknown function and the condition of its existence is formulated.