Example 1 Let $k \in \mathbf{N}_{+}, I_{k}=(2 k-1,2 k+1]$. Find the range of $a$ such that the equation $(x-2 k)^{2}=a x$ has two distinct real roots in $I_{k}$.
Solution: Rearrange the equation $(x-2 k)^{2}=a x$, we get
$$
x^{2}-(4 k+a) x+4 k^{2}=0 \text {. }
$$

Since the equation has two distinct real roots in $I_{k}$, there is the system of inequalities
$$
\left\{\begin{array}{l}
\Delta=(4 k+a)^{2}-16 k^{2}>0, \\
x_{1}=\frac{4 k+a+\sqrt{\Delta}}{2} \leqslant 2 k+1, \\
2 k-10$;
From (2) we get $a \leqslant \frac{1}{2 k+1}$;
From (3) we get $a>0$.
Therefore, $0<a \leqslant \frac{1}{2 k+1}$.
Note 1: The basic idea of this problem is to first find the roots, then restrict the roots within a certain range. This method is natural and commonly used, and many problems can be solved using this method, but it requires a higher level of skill in solving inequalities.
Note 2: This problem can also be solved using a combination of numerical and graphical methods.