Abstract
The Cauchy problem for singularly perturbed linear differential equation the second order with piecewise-constant argument is considered in the article. The definition of singularly perturbed linear harmonic oscillator with piecewise-constant argument is given in the paper. The system of fundamental solutions of homogeneous singularly perturbed differential equation with piecewiseconstant argument are constructed according to the nonhomogeneous singularly perturbed differential equation with piecewise-constant argument. With the help of the system of fundamental solutions, the initial functions are constructed and their asymptotic representation are obtained. By using the reduction method, the analytical formula of the solution of singularly perturbed the initial value problem with piecewise-constant argument is obtained. In addition, the unperturbed Cauchy problem is constructed according to the singularly perturbed Cauchy problem. The solution of the unperturbed Cauchy problem is obtained. When the small parameter tends to the zero, the solution of singularly perturbed the Cauchy problem with piecewise-constant argument approaches the solution of the unperturbed Cauchy problem with piecewise-constant argument. The theorem on the passage to the limit is proved.