Abstract
The article considers a three-term fourth order differential equation with unbounded coefficients. The coefficient of the intermediate term of the equation is assumed to be a smooth and rapidly increasing function at infinity. This intermediate term, as an operator, does not obey the differential operator formed by the extreme terms of the equation. This is precisely what makes the work unique. Using functional methods, sufficient conditions are obtained for a generalized solution of the equation to exist, be unique and maximally regular. These conditions characterize the relationship between intermediate and small coefficients. The differential equation under consideration is caused by problems of practical processes of stochastic analysis, shaft oscillations, etc. The article uses innovative methods, such as obtaining an a priori estimate of the solution, reducing the problem to the problem of invertibility of a third-order differential operator with a constant potential, and constructing a pseudo-resolvent using correct local operators. In general, the article substantiates an effective method for solving the main problems posed for differential equations on an infinite interval in the case of a new equation with an unbounded interval coefficient. Although the coefficients are assumed to be smooth, the work does not impose restrictions on the change of their derivatives. This, in turn, allows us to cover a wide class of fourth-order equations. The methods developed in the work and the results obtained can be used in a qualitative study of higher-order differential equations.