If the form of the solution of the differential equation $(y^3+x) \frac{dy}{dx} = y$ with the condition $y(4) = 2$ is $y^3 = ax + b$,then $4a + 12b^2 = $

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

Let $f:[2,5] \rightarrow R$ be a differentiable function and $\frac{f(5)}{f(2)}=1$. If there is a $c \in (2,5)$ such that $c f^{\prime}(c)=2 f(c)-2 c^3$, then $f(x)=$

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

If the solution curve of the differential equation $(y-2 \ln x) dx + (x \ln x^2) dy = 0, x > 1$ passes through the points $(e, \frac{4}{3})$ and $(e^4, \alpha)$,then $\alpha$ is equal to $................$.

Solve the differential equation $\left(\tan ^{-1} y-x\right) d y=\left(1+y^{2}\right) d x$.