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

The general solution of the differential equation $\left(1+e^{\frac{x}{y}}\right) dx + \left(1-\frac{x}{y}\right) e^{\frac{x}{y}} dy = 0$ is ($C$ is an arbitrary constant).

The solution of $x \frac{dy}{dx} = y + x e^{y/x}$ with $y(1) = 0$ 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

Suppose that $f(x, y)$ and $g(x, y)$ are homogeneous functions of the same order. If $x=Vy$ reduces the equation $\frac{dy}{dx}=\frac{f(x, y)}{g(x, y)}$ to the form $\frac{dV}{dy}=\frac{1}{y}(F(V))$,then $F(V)=$

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