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The algebraic structure of the first integrals of third-order linear equations. (English) Zbl 0986.34033

Here, the authors compare the Lie point symmetries of linear third-order ordinary differential equations with the Lie point symmetries of their linear second-order first integrals.

In particular, they find that if the third-order equation has seven symmetries, then two of its first integrals have four symmetries, and one has three; if the third-order equation has five symmetries, then all three first integrals have three symmetries; if the third-order equation has four symmetries, then all three first integrals have two symmetries. The authors do not give a theoretical explanation of such findings.

Finally, the authors show two examples of nonlinear third-order equations. The equation 2y ' y ''' -3y '' 2 =0 is found to have six symmetries, while its first integrals have three, two and one symmetry, respectively. The equation (1+y ' 2 )y ''' =(3y ' +1)y '' 2 is found to have four symmetries, while one of its first integrals has three symmetries, and two have one symmetry. No general rule is provided for nonlinear equations.

MSC:
34C14Symmetries, invariants (ODE)
34A30Linear ODE and systems, general
34A25Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.)