2009 Putnam A2

Functions $f,g,h$ are differentiable on some open interval around $0$

and satisfy the equations and initial conditions

\begin{gather*}

f' = 2f^2gh+\frac{1}{gh},\quad f(0)=1, \\

g'=fg^2h+\frac{4}{fh}, \quad g(0)=1, \\

h'=3fgh^2+\frac{1}{fg}, \quad h(0)=1.

\end{gather*}

Find an explicit formula for $f(x)$, valid in some open interval around $0$.

函数 $f,g,h$ 在 $0$ 附近的某个开区间上可微

并满足方程和初始条件

\开始{收集*}

f' = 2f^2gh+\frac{1}{gh},\quad f(0)=1, \\

g'=fg^2h+\frac{4}{fh}, \quad g(0)=1, \\

h'=3fgh^2+\frac{1}{fg}, \quad h(0)=1。

\结束{聚集*}

找到 $f(x)$ 的显式公式，该公式在 $0$ 附近的某个开区间内有效。