Loewy decomposition

In the study of differential equations, the Loewy decomposition breaks every linear ordinary differential equation (ODE) into what are called largest completely reducible components. It was introduced by Alfred Loewy.

Solving differential equations is one of the most important subfields in mathematics. Of particular interest are solutions in closed form. Breaking ODEs into largest irreducible components, reduces the process of solving the original equation to solving irreducible equations of lowest possible order. This procedure is algorithmic, so that the best possible answer for solving a reducible equation is guaranteed. A detailed discussion may be found in.

Loewy's results have been extended to linear partial differential equations (PDEs) in two independent variables. In this way, algorithmic methods for solving large classes of linear pde's have become available.

Let ${\displaystyle D\equiv {\frac {d}{dx}}}$ denote the derivative w.r.t. the variable ${\displaystyle x}$. A differential operator of order ${\displaystyle n}$ is a polynomial of the form

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