The solution of the equation $\frac{dy}{dx} = \frac{y}{x} \left( \log \frac{y}{x} + 1 \right)$ is

The general solution of ${y^2}\,dx + ({x^2} - xy + {y^2})\,dy = 0$ is

Show that the family of curves for which the slope of the tangent at any point $(x, y)$ on it is $\frac{x^{2}+y^{2}}{2 x y}$ is given by $x^{2}-y^{2}=c x$.

Let $y=y(x)$ be the solution of the differential equation $x \tan \left(\frac{y}{x}\right) d y=\left(y \tan \left(\frac{y}{x}\right)-x\right) d x$ for $-1 \leq x \leq 1$,with the initial condition $y\left(\frac{1}{2}\right)=\frac{\pi}{6}$. Then the area of the region bounded by the curves $x=0$,$x=\frac{1}{\sqrt{2}}$,and $y=y(x)$ in the upper half plane is:

The solution of the differential equation $x y^{\prime} = 2 x e^{-y / x} + y$ is

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