Example 7 Find all positive solutions of $5 x_{1}+3 x_{2}=52$.

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Given $x_{1}=8, x_{2}=4$ is a particular solution, from equations (5) and (14) we know all positive solutions are:
$$\begin{array}{c}
x_{1}=8+3 t, \quad x_{2}=4-5 t \\
-2=[-8 / 3]+1 \leqslant t \leqslant-[-4 / 5]-1 \leqslant 0
\end{array}$$

Therefore, there are three sets of positive solutions: 8, 4; 5, 9; 2, 14. It is easy to see that $x_{1}=0$ or $x_{2}=0$ cannot be solutions, so these are also all the non-negative solutions.