Geodesic
Linear ODEs and $\mathcal D$-modules, solving and decomposing equations using symmetry methods
Lobachevskii journal of mathematics, Tome 17 (2005), pp. 149-212

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This text investigates homogeneous systems of linear ODEs with smooth coefficients. Associating to an equation a differential module proves that these equations form a monoidal category with respect to the tensor product of modules, and objects in this category include homomorphisms, symmetric and exterior powers as well as dual equations. Viewing symmetries as endomorphisms of the $\mathcal D$-modules enables direct application of results from the theory of representations of Lie algebras. In particular we find decomposition and solution methods of equations with semisimple symmetry algebras, as well as solvable symmetry algebras. Sufficient conditions for equations to be solved by algebraic manipulations and quadrature are given, and unlike most previous results, there is no requirement on the symmetry algebras of having dimension equal to the order of the equations, in some cases even a single symmetry is sufficient to solve an equation.
Keywords: linear ordinary differential equations, symmetry algebras, representation theory, symmetry operators. 
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     author = {C. V. Jensen},
     title = {Linear {ODEs} and $\mathcal D$-modules, solving and decomposing equations using symmetry methods},
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     url = {http://geodesic.mathdoc.fr/item/LJM_2005_17_a6/}
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C. V. Jensen. Linear ODEs and $\mathcal D$-modules, solving and decomposing equations using symmetry methods. Lobachevskii journal of mathematics, Tome 17 (2005), pp. 149-212. http://geodesic.mathdoc.fr/item/LJM_2005_17_a6/