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Question
The integrating factor of the differential equation \[x\frac{dy}{dx} - y = 2 x^2\]
Options
e−x
e−y
\[\frac{1}{x}\]
x
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Solution
\[ \Rightarrow \frac{dy}{dx} - \frac{1}{x}y = 2x\]
\[\text{ Comparing with }\frac{dy}{dx} + Py = Q,\text{ we get }\]
\[P = - \frac{1}{x} \]
\[Q = 2x\]
Now,
\[I . F . = e^{- \int\frac{1}{x}dy} \]
\[ = e^{- \log\left| x \right|} \]
\[ = e^{log\left| \frac{1}{x} \right|} \]
\[ = \frac{1}{x}\]
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