Example 4 Solve the equation $r^{3}-[x]=3$

Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.
Solution: Let $x=[x]+r, 0 \leqslant r < 1$,
i.e., $-1 < r < 0$.
From the given equation, it is clear that $x > 0$, thus $x > 1$.
If $x \geqslant 2$, then
$$
x^{3}-x=x\left(x^{2}-1\right) \geqslant 2 \times 3=6 \text {. }
$$

Equation (2) contradicts Equation (1), hence $x < 2$.
$$
\therefore 1 < x < 2, [x]=1 \text {. }
$$

The given equation simplifies to $x^{3}=4$, solving which yields
$$
x=\sqrt[3]{4} \text {. }
$$

From the above examples, it can be seen that expressing $x$ as $[x]+r$ and using the inequality $0 \leqslant r < 1$ for solving is a common method.

The following questions involve discussions on the number of solutions to equations.