Let $y=y(x)$ be the solution of the differential equation $(xy-5x^2\sqrt{1+x^2})dx+(1+x^2)dy=0$,with $y(0)=0$. Then $y(\sqrt{3})$ is equal to

The Integrating Factor of the differential equation $x \frac{dy}{dx} - y = 2x^2$ is

Suppose $f(x)$ is a differentiable real function such that $f(x) + f'(x) \le 1$ for all $x$ and $f(0)=0$. The largest possible value of $f(1)$ is

Let $y=y(t)$ be a solution of the differential equation $\frac{dy}{dt}+\alpha y=\gamma e^{-\beta t}$,where $\alpha > 0, \beta > 0$ and $\gamma > 0$. Then $\lim_{t \rightarrow \infty} y(t)$ is:

The solution of the differential equation $x^2 \frac{dy}{dx} - xy = 1$ is...

Let $y=y(x)$ be a solution of the differential equation $(x \cos x) dy + (xy \sin x + y \cos x - 1) dx = 0$,$0 < x < \frac{\pi}{2}$. If $\frac{\pi}{3} y(\frac{\pi}{3}) = \sqrt{3}$,then $|\frac{\pi}{6} y''(\frac{\pi}{6}) + 2 y'(\frac{\pi}{6})|$ is equal to $.........$.