Abstract: To a linear (system of) ODE one may associate a very simple algebraic object which is spanned (over functions) by a vector space (the space of solutions of the equation). We shall see how one may embed the action of symmetries in this picture, in a manner which incorporates results from representation theory of Lie algebras. This provides strategies for decomposing and solving equations in terms of algebraic manipulations (eigenvalue decompositions),and we will discuss the concept of model equations for semi-simple Lie algebras of symmetries.
Concrete examples and methods will be discussed, in particular we shall see that equations on the form y''+W(x)y=0 are model equations for the semi-simple algebra sl2. Solutions of this model equation provide solutions for all other equations sharing this symmetry algebra. This approach explains why one may need only a single symmetry to solve an equation of arbitrarly high order, the success of the method depending only on eigenvalues of the action.
Relations between equations will be discussed, and the meaning of symmetries in different frameworks: One may approach this geometrically through jets, via a pure algebraic view, or, suitable for calculations, via classes of linear differential operators.