We consider canonical transformation for transforming scalar differential
equations to matrix differential equations. We determine conditions for linear
independence of solutions using the Wronskian method and use the Jordan canonical
form to find bounds for solutions of ODES. Also considered are: the generalized
eigenvectors method for obtaining matrix solutions to ODES and corresponding
bounds for the autonomous differential equations, upper and lower bounds for
solutions. Conditions for continuous dependence of solutions on initial data are
formulated. Periodic systems are studied too with the application of the Floquet rule
to finding solutions to some linear periodic systems. The Theorem on how to construct
monodromy matrices is presented for the linear periodic systems together with some
examples.
Keywords: Autonomous differential equations, Canonical transformation, Floquet rule, Jordan canonical forms, Matrix solutions, Monodromy matrices, ODES solutions, Periodic systems, Upper and lower bounds, Wronskian method.
