If $\sin x$ is the integrating factor of the linear differential equation $\frac{dy}{dx} + Py = Q$,then $P$ is

The solution of $(x+y+1) \frac{dy}{dx} = 1$ 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 $x \frac{dy}{dx} + y = x^3y^6$ is:

The integrating factor of $\left(x+2 y^3\right) \frac{d y}{d x}=y^2$ is

The solution of $\frac{dy}{dx} + 2y \tan x = \sin x$ is