If the integrating factor of $x(1 - x^2)dy + (2x^2y - y - ax^3)dx = 0$ is $e^{\int Pdx}$,then $P$ is equal to

Let $y=y(x)$ be the solution of the differential equation $2 \cos x \frac{d y}{d x}=\sin 2 x-4 y \sin x$,where $x \in \left(0, \frac{\pi}{2}\right)$. If $y\left(\frac{\pi}{3}\right)=0$,then $y^{\prime}\left(\frac{\pi}{4}\right)+y\left(\frac{\pi}{4}\right)$ is equal to:

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

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

The solution of the differential equation $x \cos x \frac{dy}{dx} + (x \sin x + \cos x) y = 1$ is

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