Given the differential equations system I need to find a general solution considering $t\gt0$ $$t\vec x '=\begin{pmatrix} 3 & -4 \\ 1 & -1 \\ \end{pmatrix} \vec x$$
A solution has the form $\vec x=\vec u t^r$ so that $(A-rI)\vec u=0$
The matrix has a repeated eigen value $\lambda=1$ so one solution is $$\vec x =\begin{pmatrix} 2 \\ 1 \\ \end{pmatrix} t$$
But now in the problem of finding another solution wich is not a multiple of the one I have found I do not know how to proceed, wich form shall the solution have as I suppose from other problems it should be something like $$\vec x=\begin{pmatrix} 2 \\ 1 \\ \end{pmatrix} tg(t)+ \begin{pmatrix} 1 \\ 0 \\ \end{pmatrix} t$$
Where $(A-I)\begin{pmatrix} 1 \\ 0 \\ \end{pmatrix}=\begin{pmatrix} 2 \\ 1 \\ \end{pmatrix} $
So how can I find another solution for the system?