In a single throw of two dice,what is the probability of obtaining a sum of numbers greater than $7$,given that $4$ appears on the first die?

The solution of $\frac{dy}{dx} = \frac{y}{x} + \tan \frac{y}{x}$ is

$A$ curve passes through the point $\left( 1, \frac{\pi}{4} \right)$ and its slope at any point is given by $\frac{dy}{dx} = \frac{y}{x} - \cos^2 \left( \frac{y}{x} \right)$. Find the equation of the curve.

The equation of a curve passing through $(1, 0)$ for which the product of the abscissa of a point $P$ and the intercept made by a normal at $P$ on the $x$-axis equals twice the square of the radius vector of the point $P$ 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

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