The solution of the differential equation $\frac{dy}{dx} = \frac{x-2y+1}{2x-4y}$ is

The function $f(x)$ satisfies the differential equation $f^2(x) + 4f'(x)f(x) + [f'(x)]^2 = 0$. Find the general solution for $f(x)$,where $c$ is an arbitrary constant.

Find the general solution of the differential equation: $e^{x} \tan y \, dx + (1 - e^{x}) \sec^{2} y \, dy = 0$.

The general solution of the differential equation $(x-(x+y) \log (x+y)) dx + x dy = 0$ is

If the curve $y = f(x)$ passes through the point $(1, e)$ and satisfies the differential equation $dy = y(2 + \log_e x) dx, x > 0$,then $f(e)$ is equal to:

The equation of the curve passing through the point $(0, -2)$ given that at any point $(x, y)$ on the curve,the product of the slope of its tangent and the $y$-coordinate of the point is equal to the $x$-coordinate of the point,is