A method of finding an asymptotic approximation of the solution of a general class of linear evolutional time-varying systems is developed and applied to illustrative problems. including ordinary and partial differential evolutional systems, The evolutional problem is first forced into the singularly perturbed form by coordinate re-scaling. Singular perturbation methods are then used to find an approximate solution of the forced evolutional problem. The developed approach can be used to: find an approximate solution, often in analytical form, for a broad class of ordinary and partial differential equations: analyze steady-state and transient behavior of the ODE and PDE models develop hybrid asymptotic-numerical methods for solving time-varying evolutional problems.