The integrating factor of the differential equation $x \frac{dy}{dx} + y \log x = x^2$ is

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:

If $y=y(x)$ is a particular solution of $\sqrt{1-x^2} \frac{dy}{dx} + \frac{2x}{\sqrt{1-x^2}} y = x$ with $y(0)=1$,then $y\left(\frac{1}{2}\right) = $

Let $f$ be a differentiable function with $\lim _{x \rightarrow \infty} f(x)=0$. If $y^{\prime}+y f^{\prime}(x)-f(x) f^{\prime}(x)=0$ and $\lim _{x \rightarrow \infty} y(x)=0$,then:

For a differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$,suppose $f^{\prime}(x)=3 f(x)+\alpha$,where $\alpha \in \mathbb{R}, f(0)=1$ and $\lim _{x \rightarrow-\infty} f(x)=7$. Then $9 f\left(-\log _{e} 3\right)$ is equal to ............

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