We suggest a constructive method for qualitative analysis of initial and many-point boundary value problems for nonautonomous systems of ordinary differential equations with polynomial matrix. Airy, Bessel, and Hermite equations and a number of applied problems, for example, the equation of gyro motion at the spin-up stage, can be reduced to such systems. We obtain efficient stability criteria for these systems. The results supplement and refine earlierknown ones. © 2010 Pleiades Publishing, Ltd.