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