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Find an implicit formula satisfied by solutions of (x2 – 4) y′ = y

Assume solutions of the equation

    \[ (x^2 - 4)y' = y \]

exist and find an implicit formula satisfied by these solutions.


This a first-order separable equation. We compute

    \begin{align*}  (x^2 - 4)y' = y && \implies && \frac{y'}{y} &= \frac{1}{x^2-4} \\  && \implies && \int \frac{1}{y} \, dy &= \int \frac{1}{x^2-4} \, dx \\  && \implies && \log |y| &= \int \frac{1}{(x-2)(x+2)} \, dx \\  && \implies && \log |y| &= \frac{1}{4} \int \frac{1}{x-2} \, dx - \frac{1}{4} \int \frac{1}{x+2} \, dx \\  && \implies && \log |y| &= \frac{1}{4} \log |x-2| - \frac{1}{4} \log |x+2| + C \\  && \implies && 4 \log |y| &= \log |x-2| - \log|x+2| + C \\  && \implies && y^4 &= C \left( \frac{x-2}{x+2} \right) \\  && \implies && y^4(x+2) &= C(x-2). \end{align*}

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