The solution of the differential equation $x \, dy - y \, dx = \sqrt{x^2 + y^2} \, dx$ is

Let $y(x)$ be the solution of the differential equation $x^2 \frac{dy}{dx} + xy = x^2 + y^2$,$x > \frac{1}{e}$,satisfying $y(1) = 0$. Then the value of $2 \frac{(y(e))^2}{y(e^2)}$ is $....$

If $(x^2+y^2) dy = xy dx$,with $y(x_0) = e$ and $y(1) = 1$,then $x_0$ has the value:

$A$ 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 $\frac{dy}{dx} = \frac{xy}{x^2 + y^2}$ is

Let $y=y(x)$ be the solution of the differential equation $x \tan \left(\frac{y}{x}\right) d y=\left(y \tan \left(\frac{y}{x}\right)-x\right) d x$ for $-1 \leq x \leq 1$,with the initial condition $y\left(\frac{1}{2}\right)=\frac{\pi}{6}$. Then the area of the region bounded by the curves $x=0$,$x=\frac{1}{\sqrt{2}}$,and $y=y(x)$ in the upper half plane is: