Question

$\frac{dy}{dx}={\sin }^{2}y$

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \frac{dy}{dx}={\sin }^{2}y \\ \Rightarrow \frac{dx}{dy}=\frac{1}{{\sin }^{2}y} \\ \Rightarrow dx={\mathrm{cosec}}^{2}ydy \\ \mathrm{Integrating}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ \int dx=\int {\mathrm{cosec}}^{2}ydy \\ \Rightarrow x=-\cot y+C \\ \Rightarrow x+\cot y=C \\ \mathrm{Hence},x+\cot y=C\mathrm{is}\mathrm{the}\mathrm{required}\mathrm{solution}. \end{gathered}$$