When $3$ dice are thrown at a time, the sum of the numbers appeared on $3$ dice was found to be $15$. Then the probability that the number $5$ does not appear on any one of the dice 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.

One ticket is selected at random from $50$ tickets numbered $00, 01, 02, \ldots, 49$. The probability that the sum of the digits is $10$,given that the product of the digits is $9$,is

The solution of the differential equation $x \frac{dy}{dx} = y(\log y - \log x + 1)$ is

The equation of the curve which passes through point $(1,0)$ and has a tangent with slope $1+\frac{y}{x}+\left(\frac{y}{x}\right)^{2}$ is

The particular solution of the differential equation $(x+y) dy + (x-y) dx = 0$ at $x=1, y=1$ is