The general solution of the differential equation $(x + y)dx + xdy = 0$ is

Find the particular solution satisfying the given condition:
$\frac{dy}{dx} - \frac{y}{x} + \csc\left(\frac{y}{x}\right) = 0$; $y = 0$ when $x = 1$.

For the differential equation $(x^2 + y^2) dy = xy dx$, it is given that $y(1) = 1$ and $y(x_0) = e$, then the value of $x_0$ is

The equation of the curve satisfying $x dy - y dx = \sqrt{x^2 - y^2} dx$ with the condition $y(1) = 0$ 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 slope of the tangent at any point $(x, y)$ on a curve $y = y(x)$ is $\frac{x^2+y^2}{2xy}$,where $x > 0$. If $y(2) = 0$,then a value of $y(8)$ is