The integrating factor of the differential equation $\frac{dy}{dx}(x \log x) + y = 4 \log x$ is

If $y+\frac{d}{d x}(x y)=x(\sin x+\log x)$,then find $y$.

If the solution curve $y=f(x)$ of the differential equation $(x^{2}-4)y^{\prime}-2xy+2x(4-x^{2})^{2}=0$ for $x>2$ passes through the point $(3, 15)$, then the local maximum value of $f$ is:

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

If the general solution of $(1+y^2) dx = (\operatorname{Tan}^{-1} y - x) dy$ is $x = f(y) + c e^{-\operatorname{Tan}^{-1} y}$, then $f(y) =$

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?