If $2 P(A) = P(B) = \frac{5}{13}$ and $P(A \mid B) = \frac{2}{5}$,then $P(A \cup B) =$ . . . . . . .

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

The solution of the differential equation $x^2 \frac{dy}{dx} = x^2 + xy + y^2$ 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 $3 x y' - 3 y + (x^2 - y^2)^{1/2} = 0$, satisfying the condition $y(1) = 1$ is

Find the particular solution satisfying the given condition: $2xy + y^2 - 2x^2 \frac{dy}{dx} = 0$; $y = 2$ when $x = 1$.