$A$ die is thrown twice. If the sum of the numbers obtained is $6$,what is the probability that the number $4$ appears at least once?

Find the particular solution satisfying the given condition: $\left[x \sin ^{2}\left(\frac{y}{x}\right)-y\right] d x+x d y=0$; $y=\frac{\pi}{4}$ when $x=1$.

If the solution curve $y=y(x)$ of the differential equation $y^{2} dx + (x^{2} - xy + y^{2}) dy = 0$ passes through the point $(1, 1)$ and intersects the line $y = \sqrt{3}x$ at the point $(\alpha, \sqrt{3}\alpha)$,then the value of $\log_{e}(\sqrt{3}\alpha)$ 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?

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

Consider the differential equation $\frac{dy}{dx} = \frac{y^3}{2(xy^2 - x^2)}$.
Statement $-1:$ The substitution $z = y^2$ transforms the above equation into a first-order homogeneous differential equation.
Statement $-2:$ The solution of this differential equation is $y^2 e^{-y^2/x} = C$.