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# Particular solution to the Euler-Cauchy equation with polynomial non-homegeneities

• The Euler-Cauchy differential equation is one of the first, and simplest, forms of a higher order non-constant coefficient ordinary di erential equation that is encountered in an undergraduate differential equations course. For a non-homogeneous Euler-Cauchy equation, the particular solution is typically determined by either using the method of variation of parameters or transforming the equation to a constant-coefficient equation and applying the method of undetermined coefficients. This paper demonstrates the surprising form of the particular solution for the most general n$^(th)$ order Euler-Cauchy equation when the non-homogeneity is a polynomial. In addition, a formula that can be used to compute the unknown coecients in the form of the particular solution is presented.
Mathematics Subject Classification: Primary: 34-01; Secondary: 34A05.

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