Question

The solution of the differential equation $\left(\frac{dy}{dx}\right)+{\sin }^{2}y=0$ is

• A. $y+2\cos y=c$
• B. $y-2\sin y=c$
• C. $x=\cot \left(y\right)+c$
• D. $y=\cot \left(x\right)+c$

Answer 1

The correct option is C

$x=\cot \left(y\right)+c$

Explanation for the correct option

Given: $\left(\frac{dy}{dx}\right)+{\sin }^{2}y=0$

$$\begin{gathered} \Rightarrow \frac{dy}{dx}=-{\sin }^2\left(y\right) \\ \Rightarrow \frac{dy}{-{\sin }^2\left(y\right)}=dx \end{gathered}$$

integrating both sides

$$\begin{gathered} \Rightarrow \int -{\csc }^2\left(y\right)dy=\int dx \\ \Rightarrow \cot \left(y\right)+c=x \end{gathered}$$

Hence, option C is correct.