The integrating factor of $\left(x+2 y^3\right) \frac{d y}{d x}=y^2$ is

If $\cos x \frac{dy}{dx} - y \sin x = 6x$ for $0 < x < \frac{\pi}{2}$ and $y(\frac{\pi}{3}) = 0$,then $y(\frac{\pi}{6})$ is equal to:

The general solution of the differential equation $\frac{d^2 y}{d x^2}+8 \frac{d y}{d x}+16 y=0$ 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 $.........$.

The solution of $\cos x \frac{dy}{dx} + y \sin x = 1$ is

The equation of one of the curves whose slope at any point is equal to $y+2x$ is