Example 5 Solve the equation
$$
x \sqrt{4-y^{2}}-y \sqrt{9-z^{2}}-z \sqrt{9-x^{2}}=11 .
$$
By the two-variable mean inequality, we have
$$
\begin{array}{l}
x \sqrt{4-y^{2}}-y \sqrt{9-z^{2}}-z \sqrt{9-x^{2}} \\
\leqslant \frac{1}{2}\left[x^{2}+4-y^{2}+(-y)^{2}+9-z^{2}+(-z)^{2}+9-x^{2}\right] \\
=11,
\end{array}
$$

Equality holds if and only if $\left\{\begin{array}{l}x=\sqrt{4-y^{2}}, \\ -y=\sqrt{9-z^{2}} \\ -z=\sqrt{9-x^{2}}\end{array}\right.$, 
Solving this, we get $x=\sqrt{2}, y=-\sqrt{2}, z=-\sqrt{7}$.
Upon verification, these values satisfy the original equation.