Abstract A system of quasi-linear partial differential equations describing the dynamic behavior of a flexible string in a rotational motion is obtained. To define frequency responses of a string the obtained system is linearized and then transformed to a two-point boundary-value problem. To solve the latter shooting methods are used. The paper develops an efficient original numerical algorithm based on the application of Taylor calculus and allowing for reduction of computational expenses, characteristic for traditional methods of a numerical solution of an initial-value problem, arising when the shooting methods are realized. The given example of the calculation of string frequency responses in a rotational motion illustrates the serviceability of the suggested procedure on the whole and its efficiency.