The integrating factor of the differential equation $x \frac{dy}{dx} + 2y = x^2 \log x$ is

The solution of the differential equation $ydx - xdy + \log x dx = 0$ is

If the solution of the differential equation $x y^{\prime}=y+x^2 \sin x$ subject to the condition $y(\pi)=0$ is $y=f(x)$ and $f(x)$ has an extreme value at $x=\alpha$, then

The general solution of the differential equation $y+\cos x(\frac{dy}{dx})-\cos^2 x=0$ is

If the solution $y(x)$ of the differential equation $\sin x \frac{dy}{dx} + y \cos x = e^{2x}, x \in (0, \pi)$ satisfies $y\left(\frac{\pi}{2}\right) = 0$, then $y\left(\frac{\pi}{6}\right) = $

The solution of $y' - y = 1, y(0) = -1$ is given by $y(x) = $