The substitution $y = z^{\alpha}$ transforms the differential equation $(x^2y^2 - 1)dy + 2xy^3dx = 0$ into a homogeneous differential equation for

Let $y=y(x)$ be the solution of the differential equation $x \tan \left(\frac{y}{x}\right) d y=\left(y \tan \left(\frac{y}{x}\right)-x\right) d x$ for $-1 \leq x \leq 1$,with the initial condition $y\left(\frac{1}{2}\right)=\frac{\pi}{6}$. Then the area of the region bounded by the curves $x=0$,$x=\frac{1}{\sqrt{2}}$,and $y=y(x)$ in the upper half plane is:

The general solution of the differential equation $\frac{dy}{dx} = \frac{x+y}{x-y}$ is

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

The general solution of the differential equation $\frac{dy}{dx} = \frac{2x+y-3}{2y-x+3}$ 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?