We discuss formally rigorous method of constructing solutions of the ordinary differential equations in the problems of wave propagation through stratified inhomogeneous media. The method is based on transformation of a homogeneous differential equation to a formally inhomogeneous one, the operator of which admits exact geometrical optics approximation solutions. The inhomogeneous differential equation is conventionally reduced to Volterra integral equation, which is transformed to the canonical set of two ordinary differential equations of the first-order. We propose, for the obtained system, a rigorous method for constructing a sequence of approximations to the exact solution of the initial differential equation. The scheme proposed for constructing a sequence of approximations in solving an ordinary differential equation can be used in problems of wave propagation through stratified inhomogeneous media. The method is applicable in the presence of losses and has no restrictions on the scales of inhomogeneities. The validation of the method does not use asymptotic considerations.