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

The substitution $x=vy$ converts which one of the following differential equations into an equation solvable by the variable separable method?

Let $f(x) = \sqrt{\lim_{r \rightarrow x} \left\{ \frac{2r^2 \left[(f(r))^2 - f(x)f(r)\right]}{r^2 - x^2} - r^3 e^{\frac{f(r)}{r}} \right\}}$ be differentiable in $(-\infty, 0) \cup (0, \infty)$ and $f(1) = 1$. Then the value of $ea$,such that $f(a) = 0$,is equal to:

Let $y = y(x)$ be the solution of the differential equation $x \sin(\frac{y}{x}) dy = (y \sin(\frac{y}{x}) - x) dx$, $y(1) = \frac{\pi}{2}$ and let $\alpha = \cos(\frac{e^{12}}{e^{12}})$. Then the number of integral values of $p$, for which the equation $x^2 + y^2 - 2px + 2py + \alpha + 2 = 0$ represents a circle of radius $r \leq 6$, is . . . . . . .

The solution curve of the differential equation $y \frac{dx}{dy} = x(\log_e x - \log_e y + 1)$,$x > 0, y > 0$ passing through the point $(e, 1)$ is

The solution of the differential equation $x dy - y dx = \sqrt{x^2+y^2} dx$, given that $y=1$ when $x=\sqrt{3}$, is