Systems of Differential Equations · Hard problem

A repeated eigenvalue gives $\mathbf x'=\begin{pmatrix}1&1\\-1&3\end{pmatrix}\mathbf x$ with $\mathbf x(0)=(1,0)$ the solution $x_1(t)=(1-t)e^{2t}$. At the time $x_1(t)$ is largest (for $t\ge0$), find the value of $x_2(t)$ at that same moment.