Solution.
Cubing both sides yields
\begin{gathered}
(13x+37)
-3\sqrt[3]{13x+37}\cdot\sqrt[3]{13x-37}\left(\sqrt[3]{13x+37}-\sqrt[3]{13x-37}\right)
- (13x-37)=2 \\
\Longrightarrow 74 -3\sqrt[3]{13x+37}\cdot\sqrt[3]{13x-37}(\sqrt[3]{2})
= 2 \\
\Longrightarrow (\sqrt[3]{2})\left(\sqrt[3]{169x^2-1369}\right) = 24
\end{gathered}
Cubing once more gives $2(169x^2-1369) = 13,824 \Longrightarrow 169x^2
- 8281 = 0$, so that $x=\pm 7$. Both of these solutions work in the
original equation.