## Analytic solution of transcendental equations

A decomposition technique of the solution of an *n*-th order linear differential equation into a set of solutions of 2-nd order linear differential equations is presented.

A decomposition technique of the solution of an *n*-th order linear differential equation into a set of solutions of 2-nd order linear differential equations is presented.

The paper concerns the problem of decomposition of a large-scale linear dynamic system into two subsystems. An equivalent problem is to split the characteristic polynomial of the original system into two polynomials of lower degrees. Conditions are found concerning the coefficients of the original polynomial which must be fulfilled for its factorization. It is proved that knowledge of only one of the symmetric functions of those polynomials of lower degrees is sufficient for factorization of the characteristic polynomial of the original large-scale system. An algorithm for finding all the coefficients of the decomposed polynomials and a general condition of factorization are given. Examples of splitting the polynomials of the fifth and sixth degrees are discussed in detail.

The maximal value of the error is the most important criterion in system design. It is also the most difficult one. For that reason there exist many other criteria. The extreme value of the error represents the attainable accuracy which can be obtained and the corresponding extreme time gives information about how fast the transients are. The extreme values of the error and the corresponding time are treated here as functions of the roots of the characteristic equation. The proposed analytical formulae allow designing systems with prescribed dynamic properties.

In the paper the extremal dynamic error x(τ) and the moment of time τ are considered. The extremal value of dynamic error gives information about accuracy of the system. The time τ gives information about velocity of transient. The analytical formulae enable design of the system with prescribed properties. These formulae are calculated due to the assumption that x(τ) is a function of the roots s_{1}, ..., s_{n} of the characteristic equation.

In the article solution of the problem of extremal value of x(τ) is presented, for the n-th order linear systems. The extremum of x(τ) is considered as a function of the roots s_{1}, s_{2}, ... s_{n} of the characteristic equation. The obtained results give a possibility of decomposition of the whole n-th order system into a set of 2-nd order systems.