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

If the solution curve $y=y(x)$ of the differential equation $y^{2} dx + (x^{2} - xy + y^{2}) dy = 0$ passes through the point $(1, 1)$ and intersects the line $y = \sqrt{3}x$ at the point $(\alpha, \sqrt{3}\alpha)$,then the value of $\log_{e}(\sqrt{3}\alpha)$ is equal to

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

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