Example 1 Find the positive integer solutions of the equation $2 x^{2}+5 y^{2}=11(x y-11)$.
Solution: The equation is $(2 x-y)(x-5 y)=-121$
$$
\therefore\left\{\begin{array}{l}
2 x-y=-121,-11,-1,1,11,121, \\
x-5 y=1,11,121,-121,-11,-1 .
\end{array}\right.
$$

Only one set $\left\{\begin{array}{l}2 x-y=1, \\ x-5 y=-121\end{array}\right.$ has a solution $(x, y)=$ $(14,27)$