Noether's and Poisson's methods for solving differential equation x_s^\left( m \right) = F_s \left(t,x_k^\left( m - 2 \right) ,x_k^\left( m - 1 \right) \right)

This paper studies integration of a higher-order differential equation which can be reduced to a second-order ordinary differential equation. The solution of the second-order equation can be obtained by the Noether method and the Poisson method. Then the solution of the higher-order equation can be obtained by integrating the solution of the second-order equation.