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Question

# The solution of the differential equation cot y dx = x dy is ________________.

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Solution

## Given: coty dx = x dy $\mathrm{cot}ydx=xdy\phantom{\rule{0ex}{0ex}}⇒\frac{dx}{x}=\frac{dy}{\mathrm{cot}y}\phantom{\rule{0ex}{0ex}}⇒\mathrm{tan}ydy=\frac{dx}{x}\phantom{\rule{0ex}{0ex}}\mathrm{Integrating}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get}\phantom{\rule{0ex}{0ex}}⇒\int \mathrm{tan}ydy=\int \frac{dx}{x}\phantom{\rule{0ex}{0ex}}⇒\mathrm{log}\left|\mathrm{sec}y\right|=\mathrm{log}\left|x\right|+\mathrm{log}\left|C\right|,\mathrm{where}\mathrm{log}\left|C\right|\mathrm{is}\mathrm{arbitrary}\mathrm{constant}\phantom{\rule{0ex}{0ex}}⇒\mathrm{log}\left|\mathrm{sec}y\right|=\mathrm{log}\left|Cx\right|\phantom{\rule{0ex}{0ex}}⇒\left|\mathrm{sec}y\right|=\left|Cx\right|\phantom{\rule{0ex}{0ex}}⇒\mathrm{sec}y=±Cx\phantom{\rule{0ex}{0ex}}⇒\mathrm{sec}y=Ax,\mathrm{where}A=±C$ Hence, the solution of the differential equation coty dx = x dy is $\overline{)\mathrm{sec}y=Ax}$.

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