Which of the following is a homogeneous differential equations?

$x \sin \left(\frac{y}{x}\right) d y=\left\{y \sin \left(\frac{y}{x}\right)-x\right\} d x$

The differential equation inoption (a) can be written as

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

Clearly, $f(\lambda x, \lambda y) \neq f(x, y)$.

So, it is not a homogeneous differential equation.

Similarly, differential equations in options (b) and (c) are not homogeneous. However, the differential equation in option (d) is homogeneous as it can be written as

$\frac{d y}{d x}=\frac{y \sin \left(\frac{y}{x}\right)-x}{x \sin \left(\frac{y}{x}\right)}$ or, $\frac{d y}{d x}=\varphi(x, y)$

and, $\varphi(\lambda x, \lambda y)=\varphi(x, y)$.