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Question

# The general solution of the differential equation $\frac{dy}{dx}=cotxcoty$ is

A

$\mathrm{cos}x=c\mathrm{cos}ecy$

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B

$\mathrm{sin}x=csecy$

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C

$\mathrm{sin}x=c\mathrm{cos}y$

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D

$\mathrm{cos}x=c\mathrm{sin}y$

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Solution

## The correct option is B $\mathrm{sin}x=csecy$Find the general solution of the differential equationGiven, $\frac{dy}{dx}=cotxcoty$$⇒\frac{dy}{coty}=cotxdx\phantom{\rule{0ex}{0ex}}⇒\int \mathrm{tan}ydy=\int cotxdx\phantom{\rule{0ex}{0ex}}⇒\mathrm{log}secy=\mathrm{log}\mathrm{sin}x+\mathrm{log}c\left[\because \int \mathrm{tan}\left(x\right)dx=\mathrm{log}\left(sec\left(x\right)\right)+c\right]\left[\because \int cotxdx=\mathrm{log}\left(\mathrm{sin}\left(x\right)\right)+c\right]\phantom{\rule{0ex}{0ex}}⇒secy=c\mathrm{sin}x\phantom{\rule{0ex}{0ex}}⇒csecy=\mathrm{sin}x\phantom{\rule{0ex}{0ex}}⇒\mathrm{sin}x=csecy$Hence, option (B) is the correct answer.

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