*Mathematical Methods in the Applied Sciences*Symmetry Algebra Classification of Scalar n th Order Ordinary
Differential Equations

We obtain a complete classification of scalar *n*th order ordinary
differential equations for all subalgebras of vector fields in the real
plane. While softwares like Maple can compute invariants of a given
order; our results are for a general *n*. The *n*=1 *,*2
*,*3 cases are well-known in the literature. Further, it is known
that there are three types of *n*th order equations depending upon
the point symmetry algebra they possess, viz. first-order equations
which admit an infinite dimensional Lie algebra of point symmetries,
second-order equations possessing the maximum eight point symmetries and
higher-order, *n*≥3, admitting the maximum *n*+4 dimensional
point symmetry algebra. We show that scalar *n*th order equations
for *n>*5 do not admit maximally an *n*+3
dimensional real Lie algebra of point symmetries. Moreover, we prove
that for *n>*4 equations can admit two types of
*n*+2 dimensional real Lie algebra of point symmetries: one type
resulting in nonlinear equations which are not linearizable via a point
transformation and the second type yielding linearizable (via point
transformation) equations. Furthermore, we present the types of maximal
real *n* dimensional and higher than *n* dimensional point
symmetry algebras admissible for equations of order *n*≥4 and their
canonical forms. The types of lower dimensional point symmetry algebras
which can be admitted are shown and the equations are constructible as
well. We state the relevant results in tabular form and in theorems.

27 Jun 2023

27 Jun 2023

27 Jun 2023

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