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# Notes on Diffy Qs - Sage demos for section 3.5

Press the *Activate* buttons below to launch the Sage demonstrations.
After the demonstration launches you should be able to interact with it
to change numbers. You may have to wait a little before the graph
appears. Be patient.

## Two dimensional linear system

Plot the vector field and a solution to

\(
\left[
\begin{matrix}
x' \\ y'
\end{matrix}
\right]
=
\left[
\begin{matrix}
a & b \\
c & d
\end{matrix}
\right]
\left[
\begin{matrix}
x \\ y
\end{matrix}
\right]
\)

The eigenvalues are computed and printed, the vector field is drawn, and a solution for \(0 < t < 10\)
is drawn. You can change the matrix and the initial value of the solution.

## Two dimensional linear system with given eigenvalues

Plot the vector field and a solution to

\(
\left[
\begin{matrix}
x' \\ y'
\end{matrix}
\right]
=
P
\left[
\begin{matrix}
x \\ y
\end{matrix}
\right]
\)

where \(P\) is some matrix with eigevalues \(a \pm bi,\) or \(a\) and \(c\) (set \(b=0\) to get the two real eigenvalues).
The vector field is drawn, and a solution for \(0 < t < 10\)
is drawn.

The original code is mainly due to Ryan Burkhart.