An integrating factor of the differential equation $\left(1-x^2\right) \frac{d y}{d x}+x y=\frac{x^4}{\left(1+x^5\right)}\left(\sqrt{1-x^2}\right)^3$ is

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

Consider the differential equation $\frac{dy}{dx} = \frac{1}{ax + 4y + 7}$ and the following statements:
$A$. The given differential equation is linear in $x$.
$B$. The given differential equation is not linear in $y$.
$C$. The given differential equation is linear in $y$.
$D$. $e^{ax}$ is the integrating factor of the given differential equation.
Which one of the following options is true?

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

If $x^2 y - x^3 \frac{dy}{dx} = y^4 \cos x$,then $x^3 y^{-3}$ is equal to

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