The general solution of the differential equation $\left(1+y^2\right) d x=\left(\tan ^{-1} y-x\right) d y$ is

If $y=e^{a x}(\cos b x+\sin b x)$ satisfies the equation $\frac{d^2 y}{d x^2}-K \frac{d y}{d x}+L y=0$, then $L+b K=$

Let $f:[2,5] \rightarrow \mathbf{R}$ be a differentiatiable 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 $f:[2,5] \rightarrow \mathbf{R}$ be a differentiatiable 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 $f(x)=$