The integrating factor of the differential equation $x \frac{dy}{dx} + 2y = x^2$ is . . . . . . . $(x \neq 0)$

Let $y=f(x)$ be the solution of the differential equation $\frac{dy}{dx}+\frac{xy}{x^2-1}=\frac{x^6+4x}{\sqrt{1-x^2}}$ for $-1 < x < 1$ such that $f(0)=0$. If $6 \int_{-1/2}^{1/2} f(x) dx = 2\pi - \alpha$,then $\alpha^2$ is equal to . . . . . .

The general solution of $\frac{dy}{dx} + y f^{\prime}(x) - f(x) f^{\prime}(x) = 0$, where $y \neq f(x)$, is

The equation of the curve passing through the origin and satisfying $\left(1+x^2\right) \frac{dy}{dx} + 2xy = 4x^2$ is

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

Let $y=y(x)$ be the solution of the differential equation $\sin x \frac{dy}{dx}+y \cos x=4x, x \in(0, \pi)$. If $y\left(\frac{\pi}{2}\right)=0$,then $y\left(\frac{\pi}{6}\right)$ is equal to