- #1

karush

Gold Member

MHB

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https://mathhelpboards.com/{http%3A//faculty.sfasu.edu/judsontw/ode/html-snapshot/linear02.html

Find the general solution of each of the linear system

\begin{align*}

x' & = -3 x + 4y\\

y' & = 3x - 2y

\end{align*}

$A=\begin{pmatrix}-3&4\\ 3&-2\end{pmatrix}

=\left[\begin{array}{rr}- \lambda - 3 & 4\\3 & - \lambda - 2\end{array}\right]

=\lambda^{2} + 5 \lambda - 6 = 0

\quad \lambda_1=-5\quad \lambda_2=6$

\textit{ eigenvector:}$\left[

\begin{array}{r}1\\1\end{array}\right]$

and eigenvector:

$\left[\begin{array}{r}- \frac{4}{3}\\1\end{array}\right]$

so

$Au=\begin{pmatrix}-3&4\\ 3&-2\end{pmatrix}

\left(\begin{array}{r}1\\1\end{array}\right)

=\left(\begin{array}{r}1\\1\end{array}\right)

=\lambda_1\left(\begin{array}{r}1\\1\end{array}\right)

=e^{-5t}\left[

\begin{array}{r}1\\1\end{array}\right]$

so far... but need other GE

typos probably

tried doing it w/o matrix but :(