Let $f: R \rightarrow R$ be a continuous function which satisfies $f(x) = \int_{0}^{x} f(t) \, dt$. Then, the value of $f(\log_{e} 5)$ 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:

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

If $\cos x \frac{dy}{dx} = y \sin x - 1$, where $x \neq (2n+1) \frac{\pi}{2}, n \in Z$, is the differential equation corresponding to the curve $y = f(x)$ and $f(0) = 1$, then $f(x) =$

If $y = f(x)$ is the solution of the differential equation $\frac{dy}{dx} = (\tan x - y) \sec^2 x$,$x \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right)$,such that $y(0) = 0$,then $y\left( -\frac{\pi}{4} \right)$ is equal to

The integrating factor of the linear differential equation $\frac{dy}{dx} + P(x)y = Q(x)$ is a solution of the differential equation: