Question

If curve $\frac{dy}{dx}=\frac{y^2\cot x}{2(1-y\ln \sqrt{\sin x})}$ passes through $(\frac{\pi }{2},10)$ and $x\in (0,\pi )$ then $\left[\frac{y\left(\frac{\pi }{3}\right)}{10}\right]=$, where $[.]$ is the greatest integer function

• A. $0$
• B. $1$
• C. $-1$
• D. $10$

Answer 1

The correct option is A $0$

$\frac{dy}{dx}=\frac{y^2\cot x}{2(1-y\ln \sqrt{\sin x})}\frac{dy}{dx}-\left(y\ln \sqrt{\sin x}\right)\frac{dy}{dx}=\frac{y^2\cot x}{2}\frac{1}{y}\frac{dy}{dx}=\frac{y\cot x}{2}+\frac{1}{2}\left(\ln \sin x\right)\frac{dy}{dx}\frac{d}{dx}\left(\ln y\right)=\frac{y}{2}\frac{d}{dx}\left(\ln \sin x\right)+\frac{1}{2}\left(\ln \sin x\right)\frac{d}{dx}\left(y\right)\frac{d}{dx}\left(\ln y\right)=\frac{1}{2}\frac{d}{dx}\left(y\ln \sin x\right)\ln y=\frac{1}{2}\left(y\ln \sin x\right)+c\because \text{it passes throught}(\frac{\pi }{2},10)\therefore c=\ln 10\Rightarrow y=10(\sin x{)}^{\frac{y}{2}}\left(\frac{y}{10}\right)\le 1\left[\frac{y\left(\frac{\pi }{3}\right)}{10}\right]=0$