Singularities of Ordinary Linear Differential Equations and Integrability

The Cauchy existence theorem for solutions of ordinary analytic differential equations concerns only the local existence. Integrability requires global solutions, i.e., analytic continuation on larger domains. It is then useful to have a priori some information on the positions of the singularities of the solutions. The simplest case corresponds to linear equations: The singularities of the solutions can arise only where the coefficients of the equation are themselves singular. For an isolated singular point, one can derive the (local) general structure of the solutions from a simple algebraic study. This leads to a natural distinction between two important cases: weakly singular and strongly singular equations, according to the absence or presence of essential singularities in the local general solution (essential singularities make more difficult the calculation of the solution by efficient constructive algorithms). Fuchs’s theorem gives a simple criterion to differentiate the two cases. It also provides an efficient constructive algorithm for the weakly singular case. If the equation is completely “Fuchsian,” it is then easily integrated. The alternative case of strongly singular equations will also be discussed for second-order differential equations. The Thome method classifies these strongly singular equations, and it gives an efficient algorithm to construct formal local solutions. These turn out to be asymptotic of true solutions. Summation methods (like the Borei method) can then be used to construct these true solutions.