In this paper we have shown the correctness conditions for a fourth-order binomial differential equation with unbounded variable coefficients. The equation is degenerate since it has no lower term. The second feature of the equation is that its intermediate coefficient increases rapidly. Such equations lead to a number of mathematical problems in the theory of oscillations, viscoelastic and inelastic flows, flexural waves, etc. The existence and uniqueness of a generalized solution are proved in the paper, and estimates of the weight norms of the solution and its first derivative are given. A wide class of functions satisfies the conditions imposed on the coefficients. The coefficients in the upper term must have a growth rate at infinity not exceeding a power function. The coefficients are assumed to be smooth functions, although no restrictions are imposed on their derivatives. All conditions are imposed on each coefficient and on certain relations between them.