Question

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

Second order linear equations
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asked 2021-01-27

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

Answers (1)

2021-01-28

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.
\(y=c_1e^{m_1x}+c_2e^{m_2x}\)
\(=c_1e^{-2x}+c_2e^{3x}\)

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