$3N$ spectral problem with $N$-fold substantial characteristics

Background. The work is a continuation of the work relating to cases of two differential beams, - with one $n$-fold and accordingly with $2n$-fold characteristics. The basis of the root functions of these beams was established under arbitrary disintegrating edge conditions given in $(0,1)$. This article explores the problem of decomposition $3n$-fold continuously differentiable function across the root elements of the bundle. At interval $(0,1)$, a differential beam with three $n$-fold real characteristic roots is considered $1$, $\pm \epsilon$, where $ \epsilon >1$. At the ends of the interval, disintegrating edge conditions are specified, only one of which is assigned to the end $1$, and the remaining conditions are specified in zero. Materials and methods. New methods have been used in the construction and evaluation of the problem. With regard to the task under consideration with three n-fold characteristics, then it does not fit into the scheme of solution of previous works and is connected with overcoming of accurate constructions and calculations. Results. It is noted that there is a significant difference between the tasks we consider and the classical tasks. Conclusions. Previously, different characteristic roots of the main operator were required. The problem is solved with disintegrating edge conditions, all of which are set at the left end, except the right end (such conditions are not Storm conditions).