AN EXISTENCE THEOREM FOR THE SOLUTION OF A CLASS OF A THIRD-ORDER NONLINEAR DIFFERENTIAL EQUATION WITH POLYNOMIAL RIGHT-HAND SIDE OF THE FIFTH DEGREE IN THE DOMAIN OF ANALYTICITY
The first of Riccati’s simplest nonlinear differential equations, is a kind of scalar and matrix form, is widely used in the theory of optimal Kalman-Bucy filters for the scalar form. The matrix differential equation plays an important role in the theory of Hamiltonian systems, in problems of optimal control, of economics. The following Abel equation from this category finds application in nonlinear optics, nonlinear diffusion, nonlinear wave theory. They should be supplemented by the Painleve equations, which are directly related to the theory of evolutionary processes. A common property that unites these types of equations is the presence of movable singular points that classify these equations in a general case that are not solvable in quadratures. This circumstance also actualizes the development of an analytic approximate method of solutions of this category of equations. The class of equations considered in this paper also belongs to this category. In this paper we prove the existence theorem for the solution of the class of equations under consideration in the analyticity region based on the majorant method applied to the solution of the desired equation, which allows us to construct an analytical approximate solution and obtain an a priori estimate of the error. Theoretical results are tested by the numerical experiment.