Let $f:[1, \infty) \rightarrow \mathbb{R}$ be a differentiable function. If $6 \int_{1}^{x} f(t) dt = 3xf(x) + x^{3} - 4$ for all $x \ge 1$, then the value of $f(2) - f(3)$ is

Let $y = y(x)$ be the solution of the differential equation $\cos x \frac{dy}{dx} + 2y \sin x = \sin 2x$ for $x \in (0, \frac{\pi}{2})$. If $y(\frac{\pi}{3}) = 0$,then $y(\frac{\pi}{4})$ is equal to:

The solution of $\frac{dy}{dx} + y = e^{-x}, y(0) = 0$ 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?

If for $x \geq 0$,$y=y(x)$ is the solution of the differential equation $(x+1) dy = ((x+1)^{2} + y - 3) dx$ with $y(2) = 0$,then $y(3)$ is equal to:

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