Question

The general solution of the differential equation $\frac{d y}{d x}+\sin \left(\frac{x+y}{2}\right)=\sin \left(\frac{x-y}{2}\right)$ is

• A. $\ln \left(\cot \frac{y}{2}\right)+2 \sin x=c$
• B. $\ln \left(\cot \frac{y}{4}\right)+2 \sin \frac{x}{2}=C$
• C. $\ln \left(\tan \frac{y}{2}\right)+2 \sin x=c$
• D. $\ln \left(\tan \frac{y}{4}\right)+2 \sin \frac{x}{2}=c$

Answer 1

Answer: (D)

$\frac{d y}{d x}+\sin \left(\frac{x+y}{2}\right)=\sin \left(\frac{x-y}{2}\right) \Rightarrow \frac{d y}{d x}=-2 \cos \frac{x}{2} \sin \frac{y}{2}$
$\Rightarrow \int \operatorname{cosec} \frac{y}{2} d y=\int-2 \cos \frac{x}{2} d x \Rightarrow 2 \ln \left(\tan \frac{y}{4}\right)$
$=-4 \sin \frac{x}{2}+c^{\prime} \Rightarrow \ln \left(\tan \frac{y}{4}\right)+2 \sin \frac{x}{2}=c$