If $z = z(x)$ and $(2 + \cos x)\frac{dz}{dx} + (\sin x)z = \sin x$,$z(x) > 0$ and $z(\frac{\pi}{2}) = 3$,then $z(\frac{\pi}{3})$ equals -

Let $y=y(x)$ be the solution of the differential equation $\log _{e}\left(\frac{d y}{d x}\right)=3 x+4 y$,with $y(0)=0$. If $y\left(-\frac{2}{3} \log _{e} 2\right)=\alpha \log _{e} 2$,then the value of $\alpha$ is equal to:

The general solution of the differential equation $x y(y+2) dy + (y^3-1) dx = 0$ is

The general solution of the differential equation $\cos (x+y) \frac{dy}{dx} = 1$ is

The solution of $\tan y \frac{dy}{dx} = \sin(x+y) + \sin(x-y)$ is

The solution of the differential equation $x = 1 + xy\frac{dy}{dx} + \frac{(xy)^2}{2!}\left(\frac{dy}{dx}\right)^2 + \frac{(xy)^3}{3!}\left(\frac{dy}{dx}\right)^3 + \dots$ is