4.177. $\frac{d x}{d t}=x-2 y, \frac{d y}{d t}=x+4 y$.
Solution. We form the characteristic equation

$$
\left|\begin{array}{cc}
(1-\lambda) & -2 \\
1 & (4-\lambda)
\end{array}\right|=\lambda^{2}-5 \lambda+6=0
$$

The roots of the characteristic equation are real, distinct, and positive:

$$
\lambda_{1}=2>0, \lambda_{2}=3>0
$$

(case $1 \mathrm{~b}$) - unstable node.