Example 1 Find the positive integer solutions of the equation $3 x^{2}+7 x y-2 x-5 y-35=0$.
Solution: For $x \geqslant 3$, the left side of the equation $\geqslant 9 x+21 y-2 x$
$$
-5 y-35=7 x+16 y-35 \geqslant 3 \times 7+16-35>0
$$

Therefore, $x=1$ or $2, y=17$ or 3.