If $I_{1},$ is the integrating factor of the differential equation $x \frac{d y}{d x} - y = x^{2}$ and $I_{2}$ is the integrating factor of the differential equation $x \frac{d y}{d x} + y = x^{- 2}$ then which of the Following is/are correct

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$I_{1} I_{2} = 1$

$I_{2} = x^{2} I_{1}$

$I_{1} + I_{2} = x + \frac{1}{x} + c,$ c is a integral constant

$I_{2} > I_{1} f o r x > 1$

answer is A, B, D.

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Detailed Solution

$\frac{d y}{d x} - \frac{y}{x} = x^{2} \to I_{1} = e^{- \int \frac{1}{x} d x} = e^{- \log x} = \frac{1}{x}$
$\frac{d y}{d x} + \frac{y}{x} = \frac{1}{x^{3}} \to I_{2} = e^{\int \frac{1}{x} d x} = e^{\log x} = x$

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