The value of '$n$', such that the differential equation $x^n \frac{dy}{dx} = y(\log y - \log x + 1)$; (where $x, y \in \mathbb{R}^+$) is homogeneous, is:

1

The correct answer is Option (2) → 1 ##

A differential equation of the form $\frac{dy}{dx} = f(x, y)$ is said to be homogeneous, if $f(x, y)$ is a homogeneous function of degree 0.

Now, $x^n \frac{dy}{dx} = y \left( \log \frac{y}{x} + \log_e e \right)$

$\Rightarrow \frac{dy}{dx} = \frac{y}{x^n} \left( \log_e e \cdot \left( \frac{y}{x} \right) \right) = f(x, y); (\text{Let}).$

$f(x, y)$ will be a homogeneous function of degree 0, if $n = 1$.