Example 1 How many pairs of integer solutions $(x, y)$ does the equation $\sqrt{x}+\sqrt{y}=\sqrt{200300}$ have? (2003 Singapore Mathematical Olympiad) Analysis We use squaring to eliminate the square roots.


From the given, we have: $\sqrt{x}=\sqrt{200300}-\sqrt{y}$, hence $x=200300+y-20 \sqrt{2003 y}$.
Since 2003 is a prime number, we can set $y=2003 a^{2}(0 \leqslant a \leqslant 10)$, then $x=2003(10-a)^{2}$. Therefore, there are 11 pairs of integer solutions.