Example 3 If $x, y$ are solutions to the system of equations
$$
\left\{\begin{array}{l}
1995 x+1997 y=5989 \\
1997 x+1995 y=5987
\end{array}\right. \text {, }
$$

then $\frac{x^{3} y^{2}}{x^{2}+y^{2}}=$ . $\qquad$
Solving the given system of equations, we get
$$
\left\{\begin{array}{l}
1995 x+1997 y=1995 \times 1+1997 \times 2, \\
1997 x+1995 y=1997 \times 1+1995 \times 2
\end{array}\right.
$$

Thus, $x=1, y=2$.
Therefore, $\frac{x^{3} y^{2}}{x^{2}+y^{2}}=\frac{1^{3} \times 2^{2}}{1^{2}+2^{2}}=\frac{4}{5}$.