Explanation

(b)
Given, $\frac{dy}{dx} - \frac{y}{x} = \frac{\phi\left( \frac{y}{x} \right)}{\phi^{'}\left( \frac{y}{x} \right)}$
$\Rightarrow$ $\frac{\phi^{'}\left( \frac{y}{x} \right)\left( \frac{x\ \ dy - y\ \ dx}{x^{2}} \right)}{\phi\left( \frac{y}{x} \right)} = \frac{1}{x}\ dx$
$\Rightarrow$ $\int_{}^{}\frac{\phi^{'}\left( \frac{y}{x} \right)d\left( \frac{y}{x} \right)}{\phi\left( \frac{y}{x} \right)} = \int_{}^{}\frac{1}{x}\ dx + \log k$
$\Rightarrow$ $\log\phi\left( \frac{y}{x} \right) = \log x + \log k$
$\Rightarrow$ $\phi\left( \frac{y}{x} \right) = kx$