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

$A$ cube having an edge of length $5 \ cm$ is painted on all faces and then it is cut into equal cubes of unit volume. $A$ small cube is selected at random and it is found that at least one of its faces is painted. What is the probability that $2$ more faces of it are also painted?

$A$ bag contains $5$ red balls and $3$ green balls. $A$ ball is selected at random and not replaced. $A$ second ball is then selected. The probability of selecting one red ball and one green ball is

Let $y=y(x)$ be the solution of the differential equation $(3y^2-5x^2)y dx + 2x(x^2-y^2) dy = 0$ such that $y(1)=1$. Then $|(y(2))^3-12y(2)|$ is equal to:

If the solution curve of the differential equation $\frac{dy}{dx} = \frac{x+y-2}{x-y}$ passes through the points $(2,1)$ and $(k+1, 2)$,where $k > 0$,then:

If $y=y(x)$ is the solution of $x \frac{d y}{d x}=y+x e^{-\left(\frac{y}{x}\right)}$ with $y(1)=\log e$,then find $y(e)$.