Coefficient characterization of linear differential equations with maximal symmetries

• We characterize linear and linearizable nonlinear equations admitting maximal Lie symmetries.
• We show that such equations depend only on two arbitrary functions, and characterize their Laguerre–Forsyth form.
• Out of the n − 2 characterizing equations, only one, whose general expression is found, represents a semi-invariant.
• Expressions for the infinitesimal generators of the induced equivalence group are found for the general linear equation.

A characterization of the general linear equation in standard form admitting a maximal symmetry algebra is obtained in terms of a simple set of conditions relating the coefficients of the equation. As a consequence, it is shown that in its general form such an equation can be expressed in terms of only two arbitrary functions, and its connection with the Laguerre–Forsyth form is clarified. The characterizing conditions are also used to derive an infinite family of semi-invariants, each corresponding to an arbitrary order of the linear equation. Finally a simplifying ansatz is established, which allows an easier determination of the infinitesimal generators of the induced pseudo group of equivalence transformations, for all the three most common canonical forms of the equation.