The general solution of the differential equation $(3x^2-2xy)dy+(y^2-2xy)dx=0$ is

$I: y^{\prime}=\frac{y+x}{x} ; \quad II: y^{\prime}=\frac{x^2+y}{x^3} ; \quad III: y^{\prime}=\frac{2xy}{y^2-x^2}$
$S1$: Differential equations given by $I$ and $II$ are homogeneous differential equations.
$S2$: Differential equations given by $II$ and $III$ are homogeneous differential equations.
$S3$: Differential equations given by $I$ and $III$ are homogeneous differential equations.

The solution of $x \frac{d y}{d x} = y(\log y - \log x + 1)$ is

Show that the differential equation $2 y e^{\frac{x}{y}} dx + (y - 2 x e^{\frac{x}{y}}) dy = 0$ is homogeneous and find its particular solution,given that $x = 0$ when $y = 1$.

The solution of the equation $\frac{dy}{dx} = \frac{x + y}{x - y}$ is

Suppose that $f(x, y)$ and $g(x, y)$ are homogeneous functions of the same order. If $x=Vy$ reduces the equation $\frac{dy}{dx}=\frac{f(x, y)}{g(x, y)}$ to the form $\frac{dV}{dy}=\frac{1}{y}(F(V))$,then $F(V)=$