On Lyapunov equivalence of linear differential systems with unbounded coefficients

We show that any linear homogeneous differential system can be reduced by some linear piecewise differentiable transformation whose matrix, together with its inverse, is bounded on the half-line to a system with piecewise constant coefficients of the same growth order, and any system with a uniformly small perturbation can be reduced by this linear transformation to the same system with a piecewise constant perturbation of the same smallness.