Find the general solution of the system x' = -x + 6y, y' = x - 2y.

Question

Find the general solution of the system

[latex]\begin{aligned}&x^{\prime}=-x+6 y, \&y^{\prime}=x-2 y .\end{aligned}[/latex]

Accepted Answer

Here [latex]a_{11}=-1, a_{12}=6, a_{21}=1, a_{22}=-2[/latex], and equation (3) becomes

[latex]\begin{aligned}D &=\left|\begin{array}{ll}a_{11}-\lambda & a_{12} \a_{21} & a_{22}-\lambda\end{array} ight| \&=\left(a_{11}-\lambda ight)\left(a_{22}-\lambda ight)-a_{21} a_{12}\end{aligned}[/latex]

[latex]D=\left|\begin{array}{cc}-1-\lambda & 6 \1 & -2-\lambda\end{array} ight|=(\lambda+2)(\lambda+1)-6=\lambda^{2}+3 \lambda-4=0[/latex]

which has the roots [latex]\lambda_{1}=-4, \lambda_{2}=1[/latex]. For [latex]\lambda_{1}=-4[/latex] the system of equations (2) becomes

[latex]\begin{aligned}&\left(a_{11}-\lambda ight) \alpha+a_{12} \beta=0 \&a_{21} \alpha+\left(a_{22}-\lambda ight) \beta=0\end{aligned}[/latex]

[latex]\begin{gathered}3 \alpha_{1}+6 \beta_{1}=0 \\alpha_{1}+2 \beta_{1}=0\end{gathered}[/latex]

Ignoring the first equation because it is just a multiple of the second, select [latex]\beta_{1}[/latex] to be 1 to obtain [latex](-2,1)[/latex] as a solution of the second equation. Hence a first solution is [latex]\left(-2 e^{-4 t}, e^{-4 t} ight)[/latex]. Similarly, with [latex]\lambda_{2}=1[/latex], we obtain the equations

[latex]\begin{array}{r}-2 \alpha_{2}+6 \beta_{2}=0 \\alpha_{2}-3 \beta_{2}=0\end{array}[/latex]

which have a solution [latex]\alpha_{2}=3, \beta_{2}=1[/latex]. Thus a second solution, linearly independent of the first, is given by the pair [latex]\left(3 e^{t}, e^{t} ight)[/latex]. By Theorem 12.2.3, the general solution is given by the pair

[latex](x(t), y(t))=\left(-2 c_{1} e^{-4 t}+3 c_{2} e^{t}, c_{1} e^{-4 t}+c_{2} e^{t} ight)[/latex]

NOTE. As defined in Section 9.10, the numbers -4 and 1 are eigenvalues of the [latex]\operatorname{matrix}\left(\begin{array}{rr}-1 & 6 \ 1 & -2\end{array} ight)[/latex] with corresponding eigenvectors [latex]\left(\begin{array}{r}-2 \ 1\end{array} ight)[/latex] and [latex]\left(\begin{array}{l}3 \ 1\end{array} ight)[/latex].

Question 12.3.1: Find the general solution of the system x' = -x + 6y, y' = x...

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