Putnam 2009 A2

Functions $f,g,h$ are differentiable on some open interval around $0$ and satisfy the equations and initial conditions

$$\begin{gathered}f' = 2f^2gh+\frac{1}{gh}, f(0)=1, \\ g'=fg^2h+\frac{4}{fh}, g(0)=1, \\ h'=3fgh^2+\frac{1}{fg}, h(0)=1.\end{gathered}$$

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