Explanation

(a)
Given, $y - x\ \ \frac{dy}{dx} = a\left( y^{2} + \frac{dy}{dx} \right)$
$\Rightarrow$ $\frac{dy}{dx}(a + x) = y - ay^{2}$
$\Rightarrow$ $\int_{}^{}\left( \frac{1}{y} + \frac{a}{1 - ay} \right)dy = \int_{}^{}\frac{dx}{a + x}$
$\Rightarrow$ $\log y - \log{(1 - ay)} = \log{(a + x)} + \log c$
$\Rightarrow$ $\log y = \log{(1 - ay)(a + x)c}$
$\Rightarrow$ $y = c(1 - ay)(a + x)$