Explanation

(b)
Given equation
$\frac{dy}{dx} + \sin{\left( \frac{x + y}{2} \right) = \sin\left( \frac{x - y}{2} \right)}$
$\Rightarrow \ \frac{dy}{dx} = \sin\left( \frac{x - y}{2} \right) - \sin\left( \frac{x + y}{2} \right)$
$\Rightarrow \ \frac{dy}{dx} = - 2\sin{\left( \frac{y}{2}\ \right)\cos\left( \frac{x}{2} \right)}$
$\Rightarrow cosec\ \left( \frac{y}{2} \right)dy = - 2\cos{\left( \frac{x}{2} \right)dx}$
On integrating both sides, we get
$\int_{}^{}{cosec\ \left( \frac{y}{2} \right)dy = - \int_{}^{}{2\cos{\left( \frac{x}{2} \right)dx + c}}}$
$\Rightarrow \ \frac{\log{(\tan{\frac{y}{4})\ }}}{\frac{1}{2}} = - \frac{2\sin\left( \frac{x}{2} \right)}{\frac{1}{2}} + c$
$\Rightarrow \log{(\tan{\frac{y}{4}) = c - 2\sin\left( \frac{x}{2} \right)}}$