On the average of differential equations with two independent small parameters

摘要

Differential equations with two independent small parameters are studied. An autonomous quasi-linear second-order equation containing two independent small parameters is considered. To calculate the approximation accuracy of the rigorous solution, the concept of smallness of the scalar function are introduced. The equation describing oscillations of a solid in the vicinity of the stable equilibrium state under the action of a linear elastic force and forces of dry friction viscous friction linearly dependent on velocity and of viscous friction quadratically dependent on velocity are considered. The solution of the first approximation of the averaging method approximates the solution with the first-order smallness within a certain time interval. The results show that construction of the function on the basis of simple physical considerations is not always correct and it uses the concept of solution indices, which are difficult to calculate in the nonlinear case.