Cosider the system of differential equationsx_1'=2x_1+x_2x_2'=4x_1-x_2Convert this system to a second order differential equations and solve this second order differential equations

Consider the given system f equations as $\{\begin{array}{l}{x}_1^{\prime }=2x_1+x_2\\ x_2^{\prime }=4x_1-x_2\end{array}$
Solve the first equation $x_1^{\prime }=2x_1+x_2$ for $x_2$ and obtained as $x_2=x_1^{\prime }-2x_1$
Substitude $x_2=x_1^{\prime }-2x_1$ in the equation $x_2^{\prime }=4x_1-x_2$
$x_2^{\prime }=4x_1-x_2$
$(x_1^{\prime }-2x_1{)}^{\prime }=4x_1-(x_1^{\prime }-2x_1)$
$x_1^{″}-2x_1^{\prime }=4x_1-x_1^{\prime }+2x_1^{\prime }$
$x_1^{″}-x_1^{\prime }-6x_1=0$
Thus the second order differential equation is $x_1^{″}-x_1^{\prime }-6x_1^{\prime }=0$
Consider the equation $x_1^{″}-x_1^{\prime }-6x_1=0$
The characteristics equation of the second order differential equation is
$m^2-m-6=0$
Solve the equation $m^2-m-6=0$ as follows.
$m^2-m-6=0$
$m^2-3m+2m-6=0$
$m(m-3)+2(m-3)=0$
$(m-3)(m+2)=0$
$m-3=0$ or $m+2=0$
$m=3$ or $m=-2$
Thus, the general solution of the second order differential equations is obtained as below.
$y=c_1{e}^{m_{1}x}+c_2{e}^{m_{2}x}$
$=c_1{e}^{-2x}+c_2{e}^{3x}$