Example 3 So far, apart from calculating specific values, we have no method to solve the indeterminate equation
$$x^{2}-11 y^{2}=1, \quad x>0, y>0$$
The indeterminate equation can be transformed into
$$\begin{array}{l}
x-\sqrt{11} y=\frac{1}{x+\sqrt{11} y}, \quad x>0, y>0 \\
\frac{x}{y}-\sqrt{11}=\frac{1}{y(x+\sqrt{11} y)}, \quad x>0, y>0
\end{array}$$

This indicates that the solutions \(x, y\) of the indeterminate equation (14) provide a very precise approximation of \(\sqrt{11}\). Therefore, we can try to find the solutions of (14) from the approximate fractions of \(\sqrt{11}\) obtained from equation (9). By verification, the fractions \(10 / 3, 199 / 60, 3970 / 1197\) yield
$$x=10, y=3 ; \quad x=199, y=60 ; \quad x=3970, y=1197$$

as solutions to (14), while the others are not. Of course, more solutions can be found from the approximate fractions obtained by continuing the calculations from equation (8).