Abstract
(1) (x-a)n(x-b)nDny = Ay, (a4 b), has been solved previously by Halphen (see Kamke, Differentialgleichungen, p. 541) by means of the substitution y = (x-b)n-)xF(loga) which transforms Eq. (1) into a linear one in F with constant coefficients. We effect the solution here in a much simpler way. Also, we find the limit of the solution as b -+ a. We assume a solution of the form y = (x-a)m(x-6b)n-m,. On substituting back in Eq. (1) and applying Liebniz's rule for the differentiation of a product, we find that m must be any root (mr) of the nth order polynomial equation
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