The method of variation of parameters, credited to J.L. Lagrange, expresses each solution of the perturbed system of differential equations, $x' = A(t)x + f(t)$, in terms of the solution of the associated homogeneous system, $y' = A(t)y$. In this talk, we will give an intuitive appreciation of why this result is to be expected. Using this understanding, we will then discuss the analogue of this method, developed by V.M. Alekseev, for obtaining solutions to perturbations of nonlinear system of differential equations.
