Weird ordinary differential equation I am searching for the functions $f,g:[0,\infty) \to [0,\infty)$ which are both increasing, $f'$ is strictly increasing, $g$ is the inverse of $f$ and

$$ g(x)^2=f(x)^2\cdot g'(x)$$

My approach was to get rid of $f$ by plugging $x=g(x)$ and obtain something which contains only $g$:

$$ (g\circ g)^2(x) = x^2 \cdot g' \circ g(x)$$

but this seems to lead nowhere. Do you have some ideas?

I can't even find an example, since $f(x)=x$ doesn't work because $f'$ should be strictly increasing...
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$$
\totald{}{x}\bracks{1 \over {\rm g}\pars{x}} = -\,{1 \over \fermi^{2}\pars{x}}
\quad\imp\quad
{1 \over {\rm g}\pars{x}} = -\int{\dd x \over \fermi^{2}\pars{x}} + \mbox{a constant}.
$$
$\newcommand{\+}{^{\dagger}}%
 \newcommand{\angles}[1]{\left\langle #1 \right\rangle}%
 \newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}%
 \newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}%
 \newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}%
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 \newcommand{\expo}[1]{\,{\rm e}^{#1}\,}%
 \newcommand{\fermi}{\,{\rm f}}%
 \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,}%
 \newcommand{\half}{{1 \over 2}}%
 \newcommand{\ic}{{\rm i}}%
 \newcommand{\iff}{\Longleftrightarrow}
 \newcommand{\imp}{\Longrightarrow}%
 \newcommand{\isdiv}{\,\left.\right\vert\,}%
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$$
\totald{}{x}\bracks{1 \over {\rm g}\pars{x}} = -\,{1 \over \fermi^{2}\pars{x}}
\quad\imp\quad
{1 \over {\rm g}\pars{x}} = -\int{\dd x \over \fermi^{2}\pars{x}} + \mbox{a constant}.
$$