 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 Amari Flowers 2021-01-27 Answered

Cosider the system of differential equations
$$x_1'=2x_1+x_2$$
$$x_2'=4x_1-x_2$$
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 $$\begin{cases}x_1'=2x_1+x_2\\x_2'=4x_1-x_2\end{cases}$$
Solve the first equation $$x_1'=2x_1+x_2$$ for $$x_2$$ and obtained as $$x_2=x_1'-2x_1$$
Substitude $$x_2=x_1'-2x_1$$ in the equation $$x_2'=4x_1-x_2$$
$$x_2'=4x_1-x_2$$
$$(x_1'-2x_1)'=4x_1-(x_1'-2x_1)$$
$$x_1''-2x_1'=4x_1-x_1'+2x_1'$$
$$x_1''-x_1'-6x_1=0$$
Thus the second order differential equation is $$x_1''-x_1'-6x_1'=0$$
Consider the equation $$x_1''-x_1'-6x_1=0$$
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.
$$y=c_1e^{m_1x}+c_2e^{m_2x}$$
$$=c_1e^{-2x}+c_2e^{3x}$$