$A$ 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

Let $y=y(x)$ be the solution of the differential equation $\left((x+2) e^{\left(\frac{y+1}{x+2}\right)}+(y+1)\right) d x=(x+2) d y$ with the initial condition $y(1)=1$. If the domain of $y=y(x)$ is an open interval $(\alpha, \beta)$,then $|\alpha+\beta|$ is equal to $......$

The solution of the differential equation $y \sin \left(\frac{x}{y}\right) dx = \left\{x \sin \left(\frac{x}{y}\right) - y\right\} dy$ satisfying $y\left(\frac{\pi}{4}\right) = 1$ is

If $\sin \left(\frac{y}{x}\right)=\log |x|+\frac{\alpha}{2}$ is the solution of the differential equation $x \cos \left(\frac{y}{x}\right) \frac{d y}{d x}=y \cos \left(\frac{y}{x}\right)+x$ and $y(1)=\frac{\pi}{3}$,then $\alpha^2$ is equal to

The general solution of the differential equation $(xy + y^2) dx - (x^2 - 2xy) dy = 0$ is

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