Three fair coins are tossed. If both heads and tails appear,then the probability that exactly one head appears is:

For the differential equation,the general solution for $x \cos \left( \frac{y}{x} \right) (y dx + x dy) = y \sin \left( \frac{y}{x} \right) (x dy - y dx)$,(where $c$ is the constant of integration) is

Show that the differential equation $x^{2} \frac{dy}{dx} = x^{2} + xy - 2y^{2}$ is a homogeneous equation and find its general solution.

The equation of the curve which passes through point $(1,0)$ and has a tangent with slope $1+\frac{y}{x}+\left(\frac{y}{x}\right)^{2}$ is

Solve the differential equation $y e^{\frac{x}{y}} dx = \left( x e^{\frac{x}{y}} + y^2 \right) dy$ where $y \neq 0$.

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