- #1

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## Homework Statement

Apply the eigenvalue method to find a general solution of the given system.

[tex]x_1' = 5x_1 - 9x_2[/tex]

[tex]x_2' = 2x_1 - x_2[/tex]

## Homework Equations

(A-λI)v=0

## The Attempt at a Solution

[tex]x_1' = 5x_1 - 9x_2[/tex]

[tex]x_2' = 2x_1 - x_2[/tex]

[tex]\left[

\begin{array}{cc}

5-λ & -9\\

2 & -1-λ

\end{array}

\right]=(5-λ)(-1-λ)+18=0[/tex]

[tex]λ^2-4λ-13=0[/tex]

[tex](λ-2)^2 -9=0[/tex]

[tex]λ=2+3i,\overline{λ}=2-3i[/tex]

So I plugged λ into the matrix;

[tex]\left[

\begin{array}{cc}

3-3i & -9\\

2 & -3-3i

\end{array}

\right]

\left[

\begin{array}{cc}

a\\

b

\end{array}

\right]=(3-3i)a-9b=0

2a-(3+3i)b=0[/tex]

This is where I'm stuck...the answer is;

[tex]x_1(t)=3e^{2t}(c_1cos2t - 5c_2sin2t)[/tex]

[tex]x_2(t)=e^{2t}[(c_1+c_2)cos3t + (c_1-c_2)sin3t)][/tex]