I'm reading [this](https://www.unf.edu/~mzhan/chapter4.pdf) pdf.

In the chapter 2, the author says:

[Picture](https://i.stack.imgur.com/PiXUn.png)

Suddenly the author states he has the idea that solution can be in the form
$\textbf{x}(t)=\textbf{v}e^{\lambda t}$.
However, i don't know where this idea came from. What is the intuition ?