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

If $y = \left(\frac{2}{\pi} x - 1\right) \operatorname{cosec} x$ is the solution of the differential equation $\frac{dy}{dx} + p(x) y = \frac{2}{\pi} \operatorname{cosec} x$ for $0 < x < \frac{\pi}{2}$,then the function $p(x)$ is equal to

If $\log y$ is an integrating factor of $\frac{dx}{dy} + P(y)x = Q(y)$,then $P(y) =$

The solution of $(2y - x) \frac{dy}{dx} = 1$ is

Let $y=y(x)$ be a differentiable function in the interval $(0, \infty)$ such that $y(1)=2$ and $\lim_{t \rightarrow x} \left( \frac{t^{2}y(x)-x^{2}y(t)}{x-t} \right) = 3$ for each $x>0$. Then $2y(2)$ is equal to

The solution of the differential equation $x \frac{dy}{dx} + 2y = x^2$ $(x \neq 0)$ with $y(1) = 1$ is