The solution of the differential equation $\frac{dy}{dx} = \tan \left(\frac{y}{x}\right) + \frac{y}{x}$ is

If $\frac{dy}{dx} = \frac{xy}{x^2 + y^2}$ and $y(1) = 1$,then find the value of $x$ that satisfies $y(x) = e$.

If the solution curve of the differential equation $\frac{dy}{dx} = \frac{x+y-2}{x-y}$ passing through the point $(2,1)$ is $\tan^{-1}\left(\frac{y-1}{x-1}\right) - \frac{1}{\beta} \log_e\left(\alpha + \left(\frac{y-1}{x-1}\right)^2\right) = \log_e|x-1|$,then $5\beta + \alpha$ is equal to

The general solution of the differential equation $(x-y-1) dy = (x+y+1) dx$ is

If $\frac{dy}{dx} = \frac{y + x \tan(\frac{y}{x})}{x}$, then $\sin(\frac{y}{x})$ is equal to

$A$ 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