Explanation

(b)
$y - x\frac{dy}{dx} = a\left( \ y^{2} + \frac{dy}{dx}\ \right)$
$\Rightarrow y - ay^{2} = a\ \frac{dy}{dx} + x\ \frac{dy}{dx}$
$\Rightarrow y(1 - ay) = (a + x)\frac{dy}{dx}$
$\Rightarrow \ \frac{dx}{(a + x)} = \frac{dy}{y(1 - ay)}$
On integrating both sides, we get
$\int_{}^{}\frac{dx}{(a + x)} = \int_{}^{}\frac{dy}{y(1 - ay)}$
$\Rightarrow \log{(a + x) = \int_{}^{}{\left\lbrack \frac{1}{y} + \frac{a}{(1 - ay)} \right\rbrack dx}}$
$\Rightarrow \log{(a + x) = \log{y + \frac{a\log{(1 - ay)}}{- a} + \log c}}$
$\Rightarrow \log{(a + x) = \log{y - \log{(1 - ay) + \log c}}}$
$\Rightarrow \log{(x + a)(1 - ay) = \log{cy}}$
$\Rightarrow (x + a)(1 - ay) = cy$