Integration of the n-th order Linear Differential Equations with Coefficients with Variable Exponential Solutions

This paper covers not linear differential equations (LDE) with variable coefficients but respective Riccati type equations which play a similar role to a characteristic equation during integration of LDE with constant coefficients. We have established a certain analogy of problems of integration of LDE in quadratures with a problem of solution to algebraic equations with radicals [5,6,7,8]. Necessary and sufficient condition for existence of an eλx form solution to an LDE of the n-th order with variable coefficients has been found. At the end of this paper we give specific examples. The solutions of this method can be used in the studies of properties of thermal conductivity, hydrophobicity of composite materials, development of new technologies multilayer asphalt and three-layer wall panel of heterogeneous materials.