## High-order ODE from system of first order ODEs

In the case of a linear, first-order system with constant coefficients $$\mathbf{x}’=A \mathbf{x},$$ with $$\mathbf{x} \in \mathbf{R}^n$$, it is known that all of the components $$x_i$$ of $$\mathbf{x}$$ satisfy the same ODE of order $$n$$, called the secular equation [at least by Birkhoff and Rota]. It also known that even in the nonlinear case an ODE of high-order can be expressed as a system of first-order ODEs.

My question is: in the case of a nonlinear system of first-order ODEs, is there a method to get a high-order ODE satisfied by the components individually? Special cases, such as polynomial RHS’s are also of interest to me.

Thank you!