Matrix presentation of ode
This process is called decoupling the equations. We have two DEs \begin{align*} x_1' & = 4x_1 + x_2 \\ x_2' & = x_1 + 3x_2. \end{align*} Differentiating the first equation once, we obtain $$ x_1'' = 4x_1' + x_2'. $$ Substituting $x_2'$ using the second equation, we get $$ x_1'' = 4x_1' + x_1 + 3x_2. $$ We still need to replace $x_2$, which can be done using the first equation again $x_2 = x_1' - 4x_1$. Hence, $$ x_1'' = 4x_1' + x_1 + 3\left(x_1' - 4x_1\right)$$ or $$ x_1'' - 7x_1' + 11x_1 = 0. $$