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

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

Let $y=y(x)$ be the solution of the differential equation $e^{y} \frac{d y}{d x}-2 e^{y} \sin x+\sin x \cos ^{2} x=0$,with $y(\frac{\pi}{2})=0$. If $y(0)=\log _{e}(\alpha+\beta e^{-2})$,then $4(\alpha+\beta)$ is equal to $....$

Let $y$ be the solution of the differential equation $(1-x^{2}) dy = (xy + (x^{3}+2) \sqrt{1-x^{2}}) dx$ for $-1 < x < 1$ with $y(0)=0$. If $\int_{-\frac{1}{2}}^{\frac{1}{2}} \sqrt{1-x^{2}} y(x) dx = k$,then $k^{-1}$ is equal to:

If $y=y(x)$ is the solution of the differential equation $\frac{dy}{dx}+\frac{4x}{x^2-1}y=\frac{x+2}{(x^2-1)^{5/2}}$ for $x > 1$,such that $y(2)=\frac{2}{9}\log_e(2+\sqrt{3})$ and $y(\sqrt{2})=\alpha\log_e(\sqrt{\alpha}+\beta)+\beta-\sqrt{\gamma}$,where $\alpha, \beta, \gamma \in N$,then $\alpha\beta\gamma$ is equal to $........$.

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