The solution of the differential equation $y \frac{dy}{dx} = x \left[ \frac{y^2}{x^2} + \frac{\phi(y^2/x^2)}{\phi'(y^2/x^2)} \right]$ is (where $c$ is a constant):

The general solution of $\left(1+e^{\frac{x}{y}}\right) d x+e^{\frac{x}{y}}\left(1-\frac{x}{y}\right) d y=0$ 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 slope of the tangent at any point $(x, y)$ on a curve $y = y(x)$ is $\frac{x^2+y^2}{2xy}$,where $x > 0$. If $y(2) = 0$,then a value of $y(8)$ is

The curve satisfying the differential equation $(x^2 - y^2) \, dx + 2xy \, dy = 0$ and passing through the point $(1, 1)$ is

The solution of the equation $x\frac{dy}{dx} = y - x\tan \left( \frac{y}{x} \right)$ is