Cauchy-Euler... Repeated Roots: Why ln(x)*x^m??????? Reduction of order

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We solve the 2nd Order Cauchy-Euler equation and investigate why we can multiply our first answer by ln(x) to get our second solution. In our process, we use reduction of order, and do a fair bit of differential equation solving.
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Bravo!!! I've been looking all over for an explanation of where mysterious natural log factor comes from. Thank you for refusing to put forward yet another "hand waving" account of why we do this to solve Cauchy Euler equations.

gentlemandude
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thanks a lot, I was stuck what to do after simplifying the terms this cleared that confusion :)

radtau