Abstract
The solution space of a linear homogeneous differential equation is a vector space. Conversely, given a finite dimensional vector space of analytic functions, it is easy to construct (see below) a differential equation with the given vector space as its solution space. The theorem proved below gives two properties of a finite dimensional vector space of analytic functions, each of which is a necessary and sufficient condition for the corresponding differential equation to be a constant coefficient differential equation. The solution space of the nth order linear homogeneous differential equation with analytic coefficients, d_n dy an(z) dz +*......al(z) + ao(z)y = O, with an(z) O,
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