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The integrating factor of the differential equation x \frac{\mathrm{d} y}{\mathrm{dx}}+2 y=\mathrm{x}^{2} is
Maths | Previous Year Question Papers
The integrating factor of the differential equation x \frac{\mathrm{d} y}{\mathrm{dx}}+2 y=\mathrm{x}^{2} is

Given x \cdot \frac{d y}{d x}+2 y=x^{2}

    \[\frac{\mathrm{dy}}{\mathrm{dx}}+\frac{2}{\mathrm{x}} \cdot \mathrm{y}=\mathrm{x}\]

This is in the form of \frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{P} y=\mathrm{Q}
here \mathrm{P}=\frac{2}{\mathrm{x}}

    \[\begin{array}{l} \text { I.F }=\mathrm{e}^{\wedge}\{\operatorname{int} \mathrm{P} \backslash \mathrm{dx}\} \$ S \\ =\mathrm{e}^{\int} \frac{2}{x} \mathrm{dx} \end{array}\]

    \[=e^{2} \int \frac{d}{x} d x\]

    \[=e^{2 . \log _{E}^{x}}\]

    \[=e^{\log z^{2}}\]

    \[=x^{2}\]

    \[\text { I.F }=x^{2}\]

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