Which of the following equations is linear?

If the solution of the differential equation $x y^{\prime}=y+x^2 \sin x$ subject to the condition $y(\pi)=0$ is $y=f(x)$ and $f(x)$ has an extreme value at $x=\alpha$,then

The general solution of the equation $\frac{dy}{dx} + \frac{1}{x}y = \frac{1}{x}e^x$ is

Let $g$ be a differentiable function such that $\int_0^x g(t) dt = x - \int_0^x tg(t) dt$ for $x \geq 0$. Let $y = y(x)$ satisfy the differential equation $\frac{dy}{dx} - y \tan x = 2(x+1) \sec x g(x)$ for $x \in [0, \frac{\pi}{2})$. If $y(0) = 0$,then $y(\frac{\pi}{3})$ is equal to

The general solution of the differential equation $e^{x} dy + (ye^{x} + 2x) dx = 0$ is

Let $f$ be a differentiable function such that $f'(x) = 7 - \frac{3}{4} \frac{f(x)}{x}$ for $x > 0$ and $f(1) \neq 4$. Then $\lim_{x \to 0^+} x f\left(\frac{1}{x}\right)$