Which of the following is a homogeneous differential equations ?

$xsin (\frac{y}{x}) dy = \begin{Bmatrix} y sin (\frac{y}{x} ) -x\end{Bmatrix}dx$

The correct answer is option (4) : $xsin (\frac{y}{x}) dy = \begin{Bmatrix} y sin (\frac{y}{x} ) -x\end{Bmatrix}dx$

The differential equation in (1) can be written as

$\frac{dy}{dx} =\frac{a^2}{(x-y)^2}$ or, $\frac{dy}{dx} = f(x, y)$

Clearly, $f (\lambda  \, x, \lambda, \, y)≠f(x, y)$

So, it is not a homogeneous differential equation.

Similarly, different equations in options (2) and (3) are not homogeneous. However, the differential equation in option (4) is homogeneous as it can be written as

$\frac{dy}{dx} = \frac{ysin (\frac{y}{x})-x}{xsin(\frac{y}{x})}$ or, $\frac{dy}{dx} = φ(x, y)$

and, $φ(\lambda \, x, \lambda \, y) = φ(x, y)$