Let $y=y(x)$ be the solution of the differential equation $x^{4}dy + (4x^{3}y + 2\sin x)dx = 0$, $x>0$, $y(\frac{\pi}{2})=0$. Then $\pi^{4}y(\frac{\pi}{3})$ is equal to:

The solution of the differential equation $x \frac{dy}{dx} + y = x^3 y^6$ is:

Let $y=y(x)$ be the solution of the differential equation $2 \cos x \frac{d y}{d x}=\sin 2 x-4 y \sin x$,where $x \in \left(0, \frac{\pi}{2}\right)$. If $y\left(\frac{\pi}{3}\right)=0$,then $y^{\prime}\left(\frac{\pi}{4}\right)+y\left(\frac{\pi}{4}\right)$ is equal to:

Let $y(x)$ be the solution of the differential equation $2 x^{2} dy + (e^{y} - 2x) dx = 0$,$x > 0$. If $y(e) = 1$,then $y(1)$ is equal to:

The solution of $(2y - x) \frac{dy}{dx} = 1$ is

If $\tan x$ is an integrating factor of the differential equation $\frac{dy}{dx} + Py = Q$, then $P$ can be