Explanation

(a)
The given equation can be written as
$\left( \frac{dx}{x} - \frac{dy}{y} \right) + \frac{(x^{2}dy - y^{2}dx)}{(x - y)^{2}} = 0$
$\Rightarrow \ \left( \frac{dx}{x} - \frac{dy}{y} \right) + \frac{\left( \frac{dy}{y^{2}} - \frac{dx}{x^{2}} \right)}{\left( \frac{1}{y} - \frac{1}{x} \right)^{2}} = 0$
$\Rightarrow \ \left( \frac{dx}{x} - \frac{dy}{y} \right) + \frac{\frac{dy}{y^{2}} - \frac{dx}{x^{2}}}{\left( \frac{1}{x} - \frac{1}{y} \right)^{2}} = 0$
On integrating both sides, we get
In $|x| - In\ |y| - \frac{1}{\left( \frac{1}{x} - \frac{1}{y} \right)} = c$
$\Rightarrow$ In $\left| \frac{x}{y} \right| - \frac{xy}{(y - x)} = c$
$\Rightarrow$ In $\left| \frac{x}{y} \right| + \frac{xy}{(x - y)} = c$