The integrating factor of the differential equation $\frac{dy}{dx}(x \log x) + y = 4 \log x$ 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

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 $f(x) =$

Solve the differential equation $\left[\frac{e^{-2 \sqrt{x}}}{\sqrt{x}}-\frac{y}{\sqrt{x}}\right] \frac{d x}{dy}=1$ where $x \neq 0$.

Let $y^{\prime}(x) + y(x) g^{\prime}(x) = g(x) g^{\prime}(x)$,$y(0) = 0$,$x \in \mathbb{R}$,where $f^{\prime}(x)$ denotes $\frac{d f(x)}{d x}$ and $g(x)$ is a given non-constant differentiable function on $\mathbb{R}$ with $g(0) = g(2) = 0$. Then the value of $y(2)$ is

The general solution of the differential equation $(x+2y^3) \frac{dy}{dx} - y = 0, y > 0$ is