Example 2 Find the positive integer solutions of the indeterminate equation $7 x+19 y=213$.
Solve: All integer solutions of the original equation obtained by the method of Example 1 are $\left\{\begin{array}{l}x=25+19 t, \\ y=2-7 t\end{array} \quad(t \in \mathbf{Z})\right.$.
$$
\text { Let }\left\{\begin{array}{l}
x=25+19 t>0, \\
y=2-7 t>0,
\end{array}\right.
$$

Solving yields $-\frac{25}{19}<t<\frac{2}{7}$.
Since $t \in \mathbf{Z}$, hence $t=0,-1$.