If $x \phi(x) = \int_{5}^{x} (3t^{2} - 2 \phi'(t)) dt$,$x > -2$,and $\phi(0) = 4$,then $\phi(2)$ is .... .

Let $y=y(x)$ be the solution of the differential equation $(1+x^2) \frac{dy}{dx} + y = e^{\tan^{-1} x}$,with $y(1)=0$. Then $y(0)$ is

The solution of the differential equation $\frac{dy}{dx} + \frac{3x^2}{1 + x^3}y = \frac{\sin^2 x}{1 + x^3}$ is

Find the general solution of the differential equation: $\cos ^{2} x \frac{d y}{d x}+y=\tan x$ where $0 \leq x < \frac{\pi}{2}$.

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 $f(x) =$

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