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

The general solution of the differential equation $\frac{d y}{d x}=\frac{3 x+y}{x-y}$ is (where $C$ is a constant of integration.)

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

One ticket is selected at random from $50$ tickets numbered $\{00, 01, 02, \ldots, 49\}$. Then the probability that the sum of the digits on the selected ticket is $8$,given that the product of these digits is zero,is

The solution of the differential equation $y^{\prime} = \frac{x^2 + y^2}{xy}$,with the initial condition $y(1) = -2$,is given by:

Two aeroplanes $I$ and $II$ bomb a target in succession. The probabilities of $I$ and $II$ scoring a hit are $0.3$ and $0.2$,respectively. The second plane will bomb only if the first misses the target. The probability that the target is hit by the second plane is