The integrating factor of the differential equation $(1-x^2) \frac{dy}{dx} + xy = kx$ for $(-1 < x < 1)$ is . . . . . . .

Integrating factor of $(x+2 y^3) \frac{d y}{d x}=y^2$ is

If for $x \geq 0$,$y=y(x)$ is the solution of the differential equation $(x+1) dy = ((x+1)^{2} + y - 3) dx$ with $y(2) = 0$,then $y(3)$ is equal to:

Let a curve $y = y(x)$ pass through the point $(3,3)$ and the area of the region under this curve,above the $x$-axis and between the abscissae $3$ and $x (>3)$ be $\left(\frac{y}{x}\right)^{3}$. If this curve also passes through the point $(\alpha, 6\sqrt{10})$ in the first quadrant,then $\alpha$ is equal to $........$

The integrating factor of $y + \frac{d}{dx}(xy) = x(\sin x + \log x)$ is

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