# A note on the transformation of the linear differential equation into a system of the first order equations

## TL;DR

This paper generalizes methods for transforming linear differential equations into first-order systems, introducing new forms that facilitate solution analysis, including a nonlinear second-order equation analogous to Riccati for third-order equations.

## Contribution

It presents a generalized transformation approach for linear differential equations, yielding new forms of N-dimensional systems and a nonlinear second-order equation for third-order cases.

## Key findings

- New transformation forms for N-dimensional systems
- Derivation of a nonlinear second-order equation for third-order equations
- Potential applications in solution analysis of differential equations

## Abstract

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.

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Source: https://tomesphere.com/paper/1803.03124