The solution of the equation $\frac{dy}{dx} = \frac{x + y}{x - y}$ 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 general solution of the differential equation $x \left( \frac{dy}{dx} \right) = y \ln \left( \frac{y}{x} \right)$ is:

Let $y=y(x)$ be the solution of the differential equation $\left((x+2) e^{\left(\frac{y+1}{x+2}\right)}+(y+1)\right) d x=(x+2) d y$ with the initial condition $y(1)=1$. If the domain of $y=y(x)$ is an open interval $(\alpha, \beta)$,then $|\alpha+\beta|$ is equal to $......$

For two events $A$ and $B$,$P(B) \neq 0$ and $P(A \mid B) = 1$,then which of the following is true?

One ticket is selected at random from $50$ tickets numbered $\{00, 01, 02, \ldots, 49\}$. Then the probability that the sum of the digits on the selected ticket is $8$,given that the product of these digits is zero,is