Example 1 Given that $f(x)$ is a linear function, and $f^{(10)}(x)$ $=1024 x+1023$, find the analytical expression of $f(x)$.
Solution: Let $f(x)=a x+b$, then $f^{(10)}(x)=a^{10}\left(x-\frac{b}{1-a}\right)+\frac{b}{1-a}$. Therefore, $a^{10}\left(x-\frac{b}{1-a}\right)+\frac{b}{1-a}=1024 x+1023$. By comparing the coefficients on both sides of the equation, we get
$$
\left\{\begin{array}{l}
a^{10}=1024, \\
-\frac{a^{10} b}{1-a}+\frac{b}{1-a}=1023 .
\end{array}\right.
$$

Solving the system of equations, we get
$$
a_{1}=2, b_{1}=1 ; a_{2}=-2, b_{2}=-3 \text {. }
$$

Therefore, the required linear functions are $f(x)=2 x+1$ or
$$
f(x)=-2 x-3
$$