Abstract
The possibilities of constructing a solution to an inhomogeneous the third-order equation have been studied, in particular the Clausen equation in the vicinity of special points x=0 and x=∞. The construction of the method for indeterminate coefficients to the construction of partial solutions of the Clausen equation is shown. These ideas are extended to constructing private solutions of the simple Clausen system near a regular feature with solutions in the form of a product of two hypergeometric Clausen functions, each of which depends on one variable. A number of properties of the product of Clausen functions constructed near these features have been proved. The construction of a common solution of the main heterogeneous system of Clausen is investigated. Four new functions have been created, representing the private solutions of the heterogeneous equation and systems like Clausen.