Consider the ODEs \begin{equation} y'=f(y), y(0)=y_0 \end{equation} and \begin{equation} y'=a f(y), y(0)=y_0 \end{equation} where $f$ is Lipschitz. Let $y_1(t)$ ($y_2(t)$) be the unique solution of the first (second) ODE. Can I conclude that $y_1(at)=y_2(t)$?
My attempt. I should verify that \begin{align} \frac{d y_1(at)}{dt} = a f(y_1(at)) \end{align} that is true if and only if \begin{align} \frac{d y_1(at)}{d(at)} = f(y_1(at)) \end{align} that is true if and only if \begin{align} \frac{d y_1(x)}{d(x)} = f(y_1(x)) \end{align} which is true because $y_1$ is a solution of the first ODE. Is this correct?
Let $z(t):=y_1(at)$. Then $z'(t)=ay_1'(at)=af(y_1(at))=af(z(t))$ and $z(0)=y_0$.
Hence $z$ is a solution of the second initial value problem. Since the solution of this initial value problem is unique, we have $z=y_2$.