A new approach for integration of the initial value problem for ordinary differential equations is suggested. The algorithm is based on approximation of the solution by a system of functions that contains orthogonal exponential polynomials.

System of alternatively orthogonalized rational functions of Jacobi type on the half line [1, ∞) is defined and its properties are established. Three subsystems of proper and mixed systems of rational functions with nice properties are presented.

We introduce sequences of functions orthogonal on a finite interval: proper orthogonal rational functions, orthogonal exponential functions, and orthogonal logarithmic functions.

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