Example 7 Find the positive integer solutions of the indeterminate equation $x^{3}-y^{3}=x y+61$.

Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
Let $(x, y)$ be a positive integer solution of the equation, then $x>y$. Let $x=y+d$, then $d$ is a positive integer, and
$$\begin{aligned}
(y+d) y+61 & =(y+d)^{3}-y^{3} \\
& =3 d y^{2}+3 y d^{2}+d^{3}
\end{aligned}$$

Thus we have
$$(3 d-1) y^{2}+d(3 d-1) y+d^{3}=61$$

Therefore,
$$\begin{array}{l}
d^{3}<61 \\
d \leqslant 3
\end{array}$$

Substituting $d=1, 2, 3$ respectively, we get
$$\begin{array}{l}
2 y^{2}+2 y+1=61 \\
5 y^{2}+10 y+8=61 \\
8 y^{2}+24 y+27=61
\end{array}$$

Only the first equation has integer solutions, and since $y$ is a positive integer, $y=5$, and thus $x=6$. Therefore, the original equation has only one positive integer solution $(x, y)=(6,5)$.