The integrating factor of $\frac{dy}{dx} + y = \frac{1+y}{x}$ is

If $x^2 y - x^3 \frac{dy}{dx} = y^4 \cos x$,then $x^3 y^{-3}$ is equal to

Let $y=y(x)$ satisfy the equation $\frac{dy}{dx}-|A|=0$,for all $x>0$,where $A=\begin{bmatrix} y & \sin x & 1 \\ 0 & -1 & 1 \\ 2 & 0 & \frac{1}{x} \end{bmatrix}$. If $y(\pi)=\pi+2$,then the value of $y\left(\frac{\pi}{2}\right)$ is:

Find the general solution of the differential equation: $x \frac{dy}{dx} + 2y = x^2 \log x$.

Let $y=y(x)$ be the solution of the differential equation $(2x \ln x) \frac{dy}{dx} + 2y = \frac{3}{x} \ln x$,$x > 0$ and $y(e^{-1}) = 0$. Then,$y(e)$ is equal to

Let $y=y(x)$ be the solution of the differential equation $\sec x \frac{dy}{dx} - 2y = 2 + 3 \sin x$, where $x \in (-\frac{\pi}{2}, \frac{\pi}{2})$ and $y(0) = -\frac{7}{4}$. Then $y(\frac{\pi}{6})$ is equal to: