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A set of linear second-order differential equations is converted into asemigroup, whose algebraic structure is used to generate many novel equations.Two independent methods that can be used to derive the equations of thesemigroup are considered, namely, the Lagrangian formalism and the Lie groupapproach. The advantages and disadvantages of each method are discussed, and itis shown that the Lagrangian formalism can be established for all equations ofthe semigroup, however, the Lie group approach is only limited to a certainsub-semigroup . The obtained results are discussed in the context of theirapplications in mathematical physics.

Lagrangian formalism and Lie group approach for commutative semigroup of differential equations