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On a family of ordinary differential equations integrable in elementary functions. (English) Zbl 1457.34004
Let $$D$$ be the differentiation operator, let the functions $$y_n(x)$$ be defined recursively $y_0(x)=\exp(x),\dots, y_n(x)=x^n Dx^{-n}y_{n-1}(x), \; n\ge 1.$ Let $$P$$ be any polynomisl of degree $$m$$. The author considers the differential equation $x^n(Dx^{-1})^nP(D)(x^{-1}D)^nx^ny(x)=ay(x) \tag{1}$ with $$a\neq0$$ and proves that (1) has a fundamental system of solutions $$\{y_n(\lambda_\nu x)\}$$, where $$\lambda_1, \dots, \lambda_{n+m}$$ are the roots of the equation $$\lambda^{2n}P(\lambda)=a$$, if all roots are different. The case of a multiple root is also treated.

##### MSC:
 34A05 Explicit solutions, first integrals of ordinary differential equations
##### Keywords:
fundamental system of solutions
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##### References:
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