Question

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

Answer 1

Given: coty dx = x dy


$$\begin{gathered} \cot ydx=xdy \\ \Rightarrow \frac{dx}{x}=\frac{dy}{\cot y} \\ \Rightarrow \tan ydy=\frac{dx}{x} \\ \mathrm{Integrating}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ \Rightarrow \int \tan ydy=\int \frac{dx}{x} \\ \Rightarrow \log \left|\sec y\right|=\log \left|x\right|+\log \left|C\right|,\mathrm{where}\log \left|C\right|\mathrm{is}\mathrm{arbitrary}\mathrm{constant} \\ \Rightarrow \log \left|\sec y\right|=\log \left|Cx\right| \\ \Rightarrow \left|\sec y\right|=\left|Cx\right| \\ \Rightarrow \sec y=\pm Cx \\ \Rightarrow \sec y=Ax,\mathrm{where}A=\pm C \end{gathered}$$


Hence, the solution of the differential equation coty dx = x dy is $\underline{\sec y=Ax}$.