Example 1 Find all positive integer pairs \((x, y)\) that satisfy the equation
\[
\begin{array}{l}
x \sqrt{y}+y \sqrt{x}-\sqrt{2003 x}-\sqrt{2003 y}+\sqrt{2003 x y} \\
=2003
\end{array}
\]

[Analysis] Based on the characteristics of the problem, we can consider starting from factoring.
Given the equation can be transformed into
$$
\begin{array}{l}
\sqrt{y}(\sqrt{x})^{2}+\left[(\sqrt{y})^{2}+\sqrt{2003} \sqrt{y}-\sqrt{2003}\right] \sqrt{x}- \\
\sqrt{2003}(\sqrt{y}+\sqrt{2003})=0 .
\end{array}
$$

Thus, $(\sqrt{y} \sqrt{x}-\sqrt{2003})(\sqrt{x}+\sqrt{y}+\sqrt{2003})=0$.
It is clear that, $\sqrt{x}+\sqrt{y}+\sqrt{2003}>0$.
Therefore, $\sqrt{x y}-\sqrt{2003}=0$
$$
\Rightarrow x y=2003 \text {. }
$$

Since 2003 is a prime number, and $x, y$ are positive integers, we have
$$
(x, y)=(2003,1),(1,2003)
$$