I am reading (the italian version of) Arnold's book on Ordinary Differential Equation. On page 29 (Chapter 1, paragraph 2: vectorial fields on the real line) problem 2 says: Given the differential equation $\dot x = v(x,t)$, where $v$ is a differentiable function, if there exists a solution $x = \varphi(t)$ which verifies the intial condition $\varphi(t_0)=x_0$, prove its unicity.
Arnold gives an indication: let $y = x-\varphi(t)$, confront $y$ with a convenient equation of the form $\dot x = kx$, $k\ne 0$.
My question is: how does the substitution $y=x-\varphi(t)$ works? Isn't $x$ equal to $\varphi(t)$? Should this substitution transform the non-authonomous equation in an authonomous one? How?