# 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

Cosider the system of differential equations
${x}_{1}^{\prime }=2{x}_{1}+{x}_{2}$
${x}_{2}^{\prime }=4{x}_{1}-{x}_{2}$
Convert this system to a second order differential equations and solve this second order differential equations

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