Explanation

(a)
Given equation is, $\frac{dy}{dx} = \frac{y}{x} + \frac{\phi\ \left( \frac{y}{x} \right)}{\phi^{'}\left( \frac{y}{x} \right)}$ ...(i)
Put $y = vx\ \Rightarrow \ \frac{dy}{dx} = v + x\frac{dv}{dx}$
Now, Eq. (i) becomes
$v + x\frac{dv}{dx} = v + \frac{\phi(v)}{\phi'(v)}$
$\Rightarrow \ \frac{\phi'(v)}{\phi(v)}dv = \frac{dx}{x}$
On integrating both sides, we get
$\int_{}^{}{\frac{\phi'(v)}{\phi(v)}dv =}\int_{}^{}{\frac{1}{x}dx}$
$\Rightarrow \log{\phi(v) = \log{x + \log k}}$
$\Rightarrow \log{\phi(v) = \log{xk}}$
$\Rightarrow \ \phi(v) = kx\ \Rightarrow \ \phi\ \left( \frac{y}{x} \right) = kx\ \ \ \left( \because v = \frac{x}{y} \right)$