If $\cos \frac{y}{x} = A \log x + C$ is the general solution of $(x \sin \frac{y}{x}) dy = (y \sin \frac{y}{x} - x) dx$,then $A =$

The solution of $x dy - y dx = \sqrt{x^2 + y^2} dx$ when $y(\sqrt{3}) = 1$ is

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

The solution of the differential equation $\frac{dy}{dx} = \frac{y}{x} + \frac{\phi(y/x)}{\phi'(y/x)}$ is

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

Let the solution curve of the differential equation $x dy = (\sqrt{x^{2}+y^{2}}+y) dx$,$x > 0$,intersect the line $x = 1$ at $y = 0$ and the line $x = 2$ at $y = \alpha$. Then the value of $\alpha$ is.