I am trying to determine the stability of the zero solution of the system
$x'=
\begin{bmatrix}
 -t & 1 \\
 1 & -t 
 \end{bmatrix}x
$

Even though, a Liapunov method can only be applied to autonomous systems I was thinking we can use a similar approach as follows:

To show that the system is asymptotically stable at the point $x\_0$ it is enough to show that it is stable and that 

$$\displaystyle \lim\_{t \to +\infty} |\phi(t,t\_0)-x\_0|=0 $$ for all $x \in U(x\_0)$ for some neighborhood $U$ of $x\_0$. 

Then define $L(x(t),y(t))=x^2+y^2$ which would be the $|| \, .||^2$. 

And if we show that $L'(x,y)<0$ along trajectories then $L(x,y)$ will be strictly decreasing continuous function and lower bounded by $0$ and therefore the limit condition mentioned above will be satisfied. 

The problem is that this is not the case at least with $L$ the way I am defining it. 
So another option is that may be I need to find another norm where this does happen. 

Does that sound like something we could do? 

I tried $L(x,y)=ax^n+by^m$ but still I can't get that $L'<0$.

Is there any other approach someone can suggest? 

Any help would be greatly appreciate it. 
Thanks!