The general solution of the differential equation $(3xy+y^2) dx + (x^2+xy) dy = 0$ is

The general solution of $y^2 dx + (x^2 - xy + y^2) dy = 0$ is:

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

Let $y=y(x)$ be the solution of the differential equation $(3y^2-5x^2)y dx + 2x(x^2-y^2) dy = 0$ such that $y(1)=1$. Then $|(y(2))^3-12y(2)|$ is equal to:

Consider the differential equation $\frac{dy}{dx} = \frac{y^3}{2(xy^2 - x^2)}$.
Statement $-1:$ The substitution $z = y^2$ transforms the above equation into a first-order homogeneous differential equation.
Statement $-2:$ The solution of this differential equation is $y^2 e^{-y^2/x} = C$.

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: