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

The solution of the differential equation $\frac{dy}{dx} - 2\frac{y}{x} = x^3$ is:

If $\tan x$ is an integrating factor of the differential equation $\frac{dy}{dx} + Py = Q$, then $P$ can be

Let $y^{\prime}(x) + y(x) g^{\prime}(x) = g(x) g^{\prime}(x)$,$y(0) = 0$,$x \in \mathbb{R}$,where $f^{\prime}(x)$ denotes $\frac{d f(x)}{d x}$ and $g(x)$ is a given non-constant differentiable function on $\mathbb{R}$ with $g(0) = g(2) = 0$. Then the value of $y(2)$ is

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

If $y=y(x)$ is the solution of the differential equation $x \frac{d y}{d x}+2 y=x e^{x}, y(1)=0$,then the local maximum value of the function $z(x)=x^{2} y(x)-e^{x}$,$x \in R$ is