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Zbl 0824.34010
Kierat, Władysław
A note on applications of the algebraic derivative to solving of some differential equations. (English)
[J] Acta Univ. Palacki. Olomuc., Fac. Rerum Nat. 110, Math. 32, 63-68 (1993). ISSN 0231-9721

By $D$ is denoted the algebraic derivative and by $s$ the differential operator in Mikusinski's field $M$. The author proves first the relation $$\text{Dexp}(c(s- a)\sp{- k})= (- k)c (s- a)\sp{- k- 1}\exp(c(s- a)\sp{- k}),\quad c\in \bbfC,\quad k\in \bbfN,$$ which allows him to solve in $M$ the equation $Dx/x= Q(s)/ P(s)$, where $P$ and $Q$ are polynomials with complex coefficients and with $\deg Q< \deg P$. The linear differential equation of order $n$ and with linear functions as coefficients can be written in $M$ in the form $- P(s) Dx+ Q(s) x= R(s)$. This allows the author to discuss the existence of particular solutions of such a differential equation and to construct them if they exist. This is a generalization of Yosida's result [{\it K. Yosida}, Proc. Jap. Acad., Ser. A 59, 1-4 (1983; Zbl 0524.34009)].
[B.Stanković (Novi Sad)]

MSC 2000:: *34A25 Analytical theory of ODE
34A30 Linear ODE and systems
44A40 Calculus of Mikusinski, etc.

Keywords: algebraic derivative; differential operator; Mikusinski's field; linear differential equation

Citations: Zbl 0524.34009

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