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Find an implicit formula satisfied by solutions of (x – 1) y′ = xy

by RoRi

Assume solutions of the equation

    \[ (x-1)y' = xy \]

exist and find an implicit formula satisfied by these solutions.

This is a separable first-order equation. We compute

    \begin{align*} (x-1)y' = xy && \implies && \frac{y'}{y} &= \frac{x}{x-1} \\ && \implies && \int \frac{1}{y} \, dy &= \int \frac{x}{x-1} \, dx \\ && \implies && \log |y| &= -\log|x-1| + x + C \\ && \implies && y &= Ce^x (x-1). \end{align*}

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