Explanation

(c)
Given differential equation is
$\frac{dy}{dx} = \frac{x - y + 3}{2(x - y) + 5}$
Put $x - y = v \Rightarrow \frac{dy}{dx} = 1 - \frac{dv}{dx}$
$\therefore$ $1 - \frac{dv}{dx} = \frac{v + 3}{2v + 5} \Rightarrow \frac{dv}{dx} = \frac{v + 2}{2v + 5\ }$
$\Rightarrow$ $\int_{}^{}\left( 2 + \frac{1}{v + 2} \right)dv = \int_{}^{}{dx}$
$\Rightarrow$ $2v + \log{(v + 2)} = x + c$
$\Rightarrow$ $2(x - y) + \log{(x - y + 2)} = x + c$