# PDE characteristics clarification of paper

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.

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