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

Let the solution curve $y = y(x)$ of the differential equation $\left[\frac{x}{\sqrt{x^{2}-y^{2}}}+e^{\frac{y}{x}}\right] x \frac{dy}{dx} = x + \left[\frac{x}{\sqrt{x^{2}-y^{2}}}+e^{\frac{y}{x}}\right] y$ pass through the points $(1, 0)$ and $(2\alpha, \alpha)$,where $\alpha > 0$. Then $\alpha$ is equal to:

The general solution of the differential equation $(x^3-y^3) dx = (x^2y - xy^2) dy$ is

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

The solution of the differential equation $y \sin \left(\frac{x}{y}\right) dx = \left\{x \sin \left(\frac{x}{y}\right) - y\right\} dy$ satisfying $y\left(\frac{\pi}{4}\right) = 1$ is

The solution of the differential equation $y \frac{dy}{dx} = x \left[ \frac{y^2}{x^2} + \frac{\phi(y^2/x^2)}{\phi'(y^2/x^2)} \right]$ is (where $c$ is a constant):