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प्रश्न
Which of the following is the integrating factor of (x log x) \[\frac{dy}{dx} + y\] = 2 log x?
पर्याय
x
ex
log x
log (log x)
उत्तर
log x
We have,
\[\left( x \log x \right)\frac{dy}{dx} + y = 2 \log x\]
Dividing both sides by (x log x) we get,
\[\frac{dy}{dx} + \frac{y}{x \log x} = 2\frac{\log x}{x \log x}\]
\[ \Rightarrow \frac{dy}{dx} + \frac{y}{x \log x} = \frac{2}{x}\]
\[ \Rightarrow \frac{dy}{dx} + \left( \frac{1}{x \log x} \right)y = \frac{2}{x}\]
\[\text{ Comparing with }\frac{dy}{dx} + Py = Q \text{ we get, }\]
\[P = \frac{1}{x \log x} \text{ and }Q = \frac{2}{x}\]
\[\text{ Now, }I . F = e^{\int P\ dx} = e^{\int\frac{1}{x\log x}dx} \]
\[ = e^{log\left( \log x \right)} \]
\[ = \log x\]
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