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

Let $y=y(x)$ be the solution of the differential equation $\sin x \frac{dy}{dx}+y \cos x=4x, x \in(0, \pi)$. If $y\left(\frac{\pi}{2}\right)=0$,then $y\left(\frac{\pi}{6}\right)$ is equal to

The integrating factor of the differential equation $x \frac{dy}{dx} + y \log x = x^2$ 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?

Find the equation of a curve passing through the point $(0,2)$ given that the sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by $5$.

Let $y = y(x)$ be the solution of the differential equation,$x\frac{dy}{dx} + y = x \ln x$,for $x > 1$. If $2y(2) = \ln 4 - 1$,then $y(e)$ is equal to