The integrating factor of the differential equation $\sin y \left(\frac{d y}{d x}\right) = \cos y (1 - x \cos y)$ is

Observe the following statements:
$I$. If $dy+2xy dx=2e^{-x^2} dx$,then $ye^{x^2}=2x+c$
$II$. If $ye^{x^2}-2x=c$,then $dx=\frac{dy}{2e^{-x^2}-2xy}$
Which of the following is a correct statement?

If the slope of the tangent of the curve at any point is equal to $-y+e^{-x}$,then the equation of the curve passing through the origin is

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 the slope of the tangent to a curve $y=f(x)$ at $(x, y)$ be given by $2 \tan x(\cos x-y)$. If the curve passes through the point $(\frac{\pi}{4}, 0)$,then the value of $\int_{0}^{\pi / 2} y \, dx$ is equal to

If $u(x)$ and $v(x)$ are two independent solutions of the differential equation $\frac{d^{2} y}{d x^{2}}+b \frac{d y}{d x}+c y=0$, then which of the following is also a solution of the given differential equation?