$f(x,y) = e^{y/x} + \tan\frac{y}{x}$ is homogeneous of degree zero.
$x\log\frac{x}{x}dx + \frac{y^2}{x} - \sin^{-1}\frac{y}{x}dy = 0$ is homogeneous
$f(x,y) = x^2 + \sin x\cos y$ is not homogeneous.
$(x^2+y^2)dx - (xy^2-y^3)dy = 0$ is a homogeneous differential equation.
Step-by-Step Solution
Key Concept: A differential equation is homogeneous if and only if $f(x, tx) = f(x, y)$ is independent of $x$ (depends only on the ratio $t$).
Question (A): $f(x,tx) = e^t + \tan^{-1}(t)$ is independent of $x$, so it is homogeneous. Question (B): $\frac{dy}{dx} = \frac{\log(y/x)}{y^2/x^2 \sin^{-1}(y/x)}$ shows $f(x,tx)$ is independent of $x$, making it homogeneous. Question (C): $f(x,y) = x^2 + \sin x\cos y$ depends on $x$ even when scaled, so it is not homogeneous. Question (D): $f(x,y) = \frac{x^2+y^2}{xy^2-y^3}$ shows $f(x,tx)$ is not independent of $f(x,tx)$ patterns, so it is not homogeneous.
Correct Answer: 1,2