The integrating factor of $x \frac{dy}{dx} + 3y = x^2$ is

Let $y=y(x)$ be the solution of the differential equation $2 \cos x \frac{d y}{d x}=\sin 2 x-4 y \sin x$,where $x \in \left(0, \frac{\pi}{2}\right)$. If $y\left(\frac{\pi}{3}\right)=0$,then $y^{\prime}\left(\frac{\pi}{4}\right)+y\left(\frac{\pi}{4}\right)$ is equal to:

If $f: R \rightarrow R$ is a differentiable function such that $f^{\prime}(x) > 2f(x)$ for all $x \in R$,and $f(0) = 1$,then:

The solution of the differential equation $x \frac{dy}{dx} + y = x^3 y^6$ is:

$A$ continuous function $f: R \rightarrow R$ satisfies the equation $f(x) = x + \int_0^x f(t) \, dt$. Which of the following options is true?

The solution of the differential equation $\frac{dy}{dx} + \frac{y}{2} \sec x = \frac{\tan x}{2y}$,where $0 \le x < \frac{\pi}{2}$,and $y(0) = 1$,is given by