The solution of $x^2 + y^2 \frac{dy}{dx} = 4$ is

The general solution of $\left(\frac{dy}{dx}\right)^{2} = 1 - x^{2} - y^{2} + x^{2}y^{2}$ 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.

The general solution of the differential equation $(y \sin x + y) \frac{dy}{dx} - \cos^2 x = 0$ is:

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

Let a curve $y=f(x)$ pass through the points $(0,5)$ and $(\log_e 2, k)$. If the curve satisfies the differential equation $2(3+y) e^{2x} dx - (7+e^{2x}) dy = 0$,then $k$ is equal to