Simple proof for uniqueness of solutions of linear ODEs? Consider the system of linear ODEs $\dot{x}(t)=Ax(t)$, $x(0)=x_0\in\mathbb{R}^n$. Does anyone know a simple proof showing that the solutions are unique that does not require resorting to more general existence/uniqueness results (e.g., those relating to the Picard iteration) nor solving for the solutions explicitly?

The general solution of the initial value problem $X'=AX$ with $X(0)=X_0$ is    $$X(t)=e^{At}X_0$$ Let $Y(t)$ be an another solution satisfying  $Y(0)=X_0$. I want to show that $X(t)=Y(t)$. Define $$Z(t)=exp(-tA)\cdot Y(t)$$ $$\qquad\frac{dZ}  {dt}=-\exp(-tA)\cdot Y(t)+\exp(-tA)\cdot Y(t) =0$$ therefore $Z(t)$ is a constant. Setting $t=0$ we have $Z(t)=X_0$ so  $Y(t)=\exp(At)X_0$ with  $Y(0)=X_0$ this completes the proof.
