$A$ function $y = f(x)$ satisfies $(x + 1)f'(x) - 2(x^2 + x)f(x) = \frac{e^{x^2}}{(x + 1)}$. If $f(0) = 5$,then $f(x)$ is:

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

If $y = f(x)$ is the solution of the differential equation $\frac{dy}{dx} = (\tan x - y) \sec^2 x$,$x \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right)$,such that $y(0) = 0$,then $y\left( -\frac{\pi}{4} \right)$ is equal to

If $\frac{dy}{dx} + y \tan x = \sin 2x$ and $y(0) = 1$,then $y(\pi)$ is equal to

Let $f$ be a differentiable function such that $x^2 f(x) - x = 4 \int_0^x t f(t) dt$ and $f(1) = \frac{2}{3}$. Then $18 f(3)$ is equal to $......$.

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?