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

Let $y=y(x)$ be the solution of the differential equation $\frac{d y}{d x}=\frac{(\tan x)+y}{\sin x(\sec x-\sin x \tan x)}$,$x \in\left(0, \frac{\pi}{2}\right)$ satisfying the condition $y\left(\frac{\pi}{4}\right)=2$. Then,$y\left(\frac{\pi}{3}\right)$ is

The general solution of the differential equation $\frac{dy}{dx} + \left(\frac{3x^2}{1+x^3}\right)y = \frac{1}{x^3+1}$ is

If $x \phi(x) = \int_{5}^{x} (3t^{2} - 2 \phi'(t)) dt$,$x > -2$,and $\phi(0) = 4$,then $\phi(2)$ is .... .

If the slope of the tangent drawn at any point $(x, y)$ on a curve is $(x+y)$,then the equation of that curve is

The general solution of the differential equation $(y^2+x+1) dy = (y+1) dx$ is