Example 8. Let the sequence be $u_{1}=2$, $u_{n+1}=\frac{n+1}{n^{2}} u_{n}^{2}$, find its general term.
Solve: First, rewrite the sequence as $\frac{u_{n+1}}{n+1} = \left(\frac{u_{n}}{n}\right)^{2}$, then let $v_{n} = \log _{2} \frac{u_{n}}{n}$ to get the geometric sequence $v_{1} = 1, v_{n+1} = 2 v_{n}$, thus
$$
\begin{aligned}
v_{n} = 2^{n-1}, & u_{n} = n \cdot 2^{2^{n-1}} \cdot \\
& \equiv 、 \mathbf{u}_{n+1} = \frac{a \mathbf{u}_{n} + \mathbf{b}}{c \mathbf{u}_{n} + \mathbf{d}^{\prime}} \text{ other }
\end{aligned}
$$