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The integrating factor of the differential equation $$\left(x^2 - 1\right) \frac{dy}{dx} + 2xy = x$$ is
$$(x^2 - 1)\frac{dy}{dx} + 2xy = x$$
$$\frac{dy}{dx} + \left( \frac{2x}{x^2 - 1} \right) y = \frac{x}{x^2 - 1}$$
$$P(x) = \frac{2x}{x^2 - 1}$$
$$\text{I.F.} = e^{\int \frac{2x}{x^2 - 1} \, dx}$$
$$\text{I.F.} = e^{\ln\vert{}x^2 - 1\vert{}} = x^2 - 1$$
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