The solution of $\frac{dy}{dx} = \left( \frac{y}{x} \right)^{1/3}$ is

The general solution of the differential equation $\tan x \tan y \, dx + \cos^2 x \operatorname{cosec}^2 y \, dy = 0$ is

The equation of the curve passing through $\left(\frac{\pi}{6}, 0\right)$ and satisfying the differential equation $(e^y+1) \cos x \, dx + e^y \sin x \, dy = 0$ is:

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

Let $f: R \rightarrow R$ be a differentiable function with $f(0)=0$. If $y=f(x)$ satisfies the differential equation $\frac{dy}{dx}=(2+5y)(5y-2)$,then the value of $\lim_{x \rightarrow -\infty} f(x)$ is:

The solution of the differential equation $x^2(y+1) \frac{dy}{dx} + y^2(x+1)^2 = 0$,given $y(1) = 2$,is