is there short proof of uniqueness of solutions of a linear two-dimensional non-autonomous system of ODE? I am looking for short and simple (accessible for an economist with calculus background) proof of uniqueness of solutions of a linear two-dimensional non-autonomous system of ODE.
 A: In a linear system $y'(t)=A(t)y(t)+b(t)$, the difference $u$ of two solutions $x,y$ is itself the solution of the homogeneous system
$$
u'(t)=y'(t)-x'(t)=A(t)(y(t)-x(t))=A(t)u(t).
$$
Now apply vector and associated matrix norms
$$
\|u'(t)\|\le \|A(t)\|\,\|u(t)\|
$$
By the Grönwall lemma, this results in the upper bound
$$
\|u(t)\|\le \exp\left(\int_{t_0}^t\|A(s)\|\,ds\right)\|u(t_0)\|.
$$
So when the two solutions are equal at $t_0$, they also have to be equal for any $t>t_0$. A similar argument goes also for $t<t_0$.

To get that bound, first consider the exact equation $d'(t)=\|A(t)\|d(t)$ which has the solution $d(t)=e^{c(t)}d(t_0)$ with $c'(t)=\|A(t)\|$, $c(t_0)=0$. Now consider the difference $h_a(t)=\|u(t)\|-(\|u(t_0)\|+a)e^{c(s)}$ for some $a>0$ under the integral identities and inequalities
\begin{align}
\|u(t)\|&=\|u(t_0)\|+\left\|\int_{t_0}^tu'(s)\,ds\right\|
\le\|u(t_0)\|+\int_{t_0}^t\left\|u'(s)\right\|\,ds
\\
&\le \|u(t_0)\|+\int_{t_0}^t\|A(s)\|\,\|u(s)\|\,ds
\\
\\
e^{c(t)}&=1+\int_{t_0}^te^{c(s)}c'(s)\,ds
\\[1em]\hline
h_a(t)=\|u(t)\|-(\|u(t_0)\|+a)e^{c(t)}&
\le-a+\int_{t_0}^t\|A(s)\|\,\Bigl(\|u(s)\|-(\|u(t_0)\|+a)e^{c(s)}\Bigr)\,ds
\\&=-a+\int_{t_0}^t\|A(s)\|\,h_a(s)\,ds.
\end{align}
From this one concludes that there can be no $t$ where the $h_a(t)\ge 0$,, as then there would be a minimal such $t$ with $h_a(t)=0$ and from the last inequality $h_a(t)\le-a<0$ would follow in contradiction.
Now as
$$
\|u(t)\|<(\|u(t_0)\|+a)e^{c(t)}
$$
for all $a>0$, it follows that in the limit 
$$
\|u(t)\|\le\|u(t_0)\|e^{c(t)}.
$$
