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In my work I've arrived at the equation: $x y'(x)+ay(x)+b y(cx-1)=0$, where $a,b,c$ are given constants ($c\ge1$). I suspect this equation has been studied before. Can anyone suggest a reference? In the special case of $c=1$, the equation reduces to $x y'(x)+ay(x)+b y(x-1)=0$ and this equation has already been studied extensively in the literature, especially on number theory.

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