Example 1. Find the solution $(x, y)$ that satisfies the equation $\frac{1}{x}-\frac{1}{y}=\frac{1}{12}$, and makes $y$ the largest positive integer.
Solving the original equation for $y$, we get
$$
y=\frac{12 x}{12-x}=-12+\frac{144}{12-x}.
$$

From this equation, it is clear that for $y$ to take its maximum value, $12-x$ must be positive and as small as possible. Also, $12-x$ should be a divisor of 144, so $12-x=1, x=11$, at which point $y=132$. Therefore, the positive integer solution to the original equation is $(x, y)=(11, 132).$