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

The general solution of the differential equation $\left(\frac{y}{x}\right) \cos \left(\frac{y}{x}\right) dx - \left[\left(\frac{x}{y}\right) \sin \left(\frac{y}{x}\right) + \cos \left(\frac{y}{x}\right)\right] dy = 0$ is:

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

$A$ coin is tossed until a head appears or until the coin has been tossed five times. If a head does not occur on the first two tosses,then the probability that the coin will be tossed $5$ times is

The solution of the differential equation $x + y\frac{dy}{dx} = 2y$ is

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