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:

Let $y=y(x)$ be the solution of the differential equation $\sec^2 x dx + (e^{2y} \tan^2 x + \tan x) dy = 0$,where $0 < x < \frac{\pi}{2}$ and $y(\frac{\pi}{4}) = 0$. If $y(\frac{\pi}{6}) = \alpha$,then $e^{8\alpha}$ is equal to:

The general solution of the differential equation $(1+y^2) dx = (\tan^{-1} y - x) dy$ is

If $x \log x \frac{dy}{dx} + y = \log x^2$ and $y(e) = 0$,then $y(e^2) = $

Let $y=y(x)$ satisfy the equation $\frac{dy}{dx}-|A|=0$,for all $x>0$,where $A=\begin{bmatrix} y & \sin x & 1 \\ 0 & -1 & 1 \\ 2 & 0 & \frac{1}{x} \end{bmatrix}$. If $y(\pi)=\pi+2$,then the value of $y\left(\frac{\pi}{2}\right)$ is:

The solution of the differential equation $\frac{dy}{dx} + y \sec^2 x = \tan x \sec^2 x$ is