PDE characteristics clarification of paper

Question

I am curious how the following circled part is found. I have been trying to figure it out for the past half hour or so and I can't find the computation that gives that number $x(t)=\sqrt{2t}$.

It shows up on page 16 of the paper here

Thank you.

Answer 1

Oh I think I figured it out. I want to solve

$\frac{dx}{dt}=\frac{x}{2t}$ with $x(2)=2$. That gives the solution.