Rooms in a hotel are numbered from $1$ to $19$. Rooms are allocated at random as guests arrive. The first guest to arrive is given a room which is a prime number. The probability that the second guest to arrive is given a room which is a prime number is

The solution of the differential equation $x y^2 d y - (x^3 + y^3) d x = 0$ is

The solution of the differential equation $\frac{dy}{dx} = \frac{y}{x} + \frac{\phi(y/x)}{\phi'(y/x)}$ is

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:

Show that the differential equation $x \frac{dy}{dx} - y + \sin \left(\frac{y}{x}\right) = 0$ is a homogeneous equation and find its solution.

The real value of $m$ for which the substitution $y = u^m$ will transform the differential equation $2x^4y \frac{dy}{dx} + y^4 = 4x^6$ into a homogeneous equation is: