Unique solutions of given an initial condition proof ODEs This is on our study guide for an upcoming test, and none of us have an idea how to answer it.


  Show that if $A(t)$ is an $n \times n$ matrix with continuous dependence on $t \in [a, b]$, then for any $\tau \in [a, b]$ and any $\xi \in \mathbb{R}^n$ the solution of $$x'=A(t)x$$ with initial condition $\phi(\tau)=\xi$ exists for all $t$ in $[a,b]$ and is unique.


Thanks in advance.
Apply the theorem of Picard-Lindelöf, first on bounded intervals then using continuation to the whole of $\Bbb R$. Note that also the matrix norm $\|A(t)\|$ is a continuous function in $t$ that is bounded on bounded intervals.
Let $\mathbf \Phi$ be a $n\times n$ invertible matrix. 
$$\dfrac{d\mathbf \Phi}{dt} = A(t) \mathbf \Phi$$
Multiplying both sides at right by $\mathbf x^{-1}$
$$\dfrac{d\mathbf \Phi}{dt} \mathbf \Phi^{-1}= A(t) \mathbf I$$

Integrating both sides with respect to $t$
$$\ln \mathbf \Phi = \int A(t) dt \\
\mathbf \Phi = \exp{\bigg(\int A(t) dt \bigg)}$$

Multiplying $\mathbf \Phi(t)$ by a constant vector gives a vector $\mathbf X(t)$ which satisfies your equation, and you can evaluate in $t=\tau$

I hope it helps. 
See Boyce, DiPrima. 2012 for more information