The solution of the differential equation $(1+x) y \,dx + (1-y) x \,dy = 0$ is

If the solution $y(x)$ of the given differential equation $(e^y+1) \cos x \, dx + e^y \sin x \, dy = 0$ passes through the point $(\frac{\pi}{2}, 0)$,then the value of $e^{y(\frac{\pi}{6})}$ is equal to ...........

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

The solution of the differential equation $\frac{dy}{dx} = x^2 + \sin 3x$ 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:

$A$ particle moves in a straight line with a velocity given by $\frac{dx}{dt} = x + 1$ (where $x$ is the distance covered). The time taken by the particle to traverse a distance of $99 \ m$ is: