\begin{align}
    & 2 \mu(t) t^2 + g(t)^2 = 2t \\
    \Leftrightarrow \ & - 2 \frac{1}{1 - t} t^2 + g(t)^2 = 2 t \\
    \Leftrightarrow \ & g(t)^2 = 2 t + 2 \frac{1}{1 - t} t^2 \\
    \Leftrightarrow \ & g(t)^2 = \frac{2t}{1 - t}.
\end{align}
