  A generalization of the already studied transformations of the linear
differential equation into a system of the first order equations is given. The
proposed transformation gives possibility to get new forms of the N-dimensional
system of first order equations that can be useful for analysis of the
solutions of the N-th order differential equations. In particular, for the
third-order linear equation the nonlinear second-order equation that plays the
same role as the Riccati equation for second-order linear equation is obtained.

Classical Analysis and ODEs