On the integrability of GL(2,ℝ)‐invariant fourth‐order ordinary differential equations.

The integrability of fourth‐order ordinary differential equations admitting gl(2,ℝ) as Lie symmetry algebra is addressed in this work. The classical Lie's method of reduction cannot be applied to solve by quadrature this kind of equations because gl(2,ℝ) is nonsolvable. In order to avoid such difficulty, a solvable structure involving the vector field identified with the equation is constructed by using the symmetry generators of gl(2,ℝ). This permits to compute a first integral of the equation by quadrature. In the aftermath, it is shown that the general solution of any GL(2,ℝ)‐invariant fourth‐order ordinary differential equation can be obtained in parametric form, involving linearly independent solutions to a related one‐parameter family of linear second‐order equations. Particular examples are also shown with the end of illustrating the presented approach. [ABSTRACT FROM AUTHOR]