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!