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Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.
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Analytic hypoellipticity of linear ordinary differential operators
This is probably not what you are looking for, but the book An Introduction to Ordinary Differential Equations by Earl A. Coddington considers existence and uniqueness of classical solutions to $Pu=0$ …