Phase portraits for real eigenvalues Problem set B Answers

1. (a) \(\lambda=2\ \text{and}\ 7\)
2. (b) \(\lambda=-6\ \text{and}\ -1\)

Unstable node: eigenvalues \(2\text{ and }3\); eigenvector lines gradient −2 (\(\lambda=2\)) and gradient −1 (\(\lambda=3\)); arrows away from the origin on both lines

Stable node: eigenvalues \(-5\text{ and }-1\); eigenvector lines gradient −3 (\(\lambda=-5\)) and gradient 1 (\(\lambda=-1\)); arrows towards the origin on both lines

Saddle point: eigenvalues \(-1\text{ and }5\); eigenvector lines gradient −2 (\(\lambda=-1\)) and gradient 1 (\(\lambda=5\)); arrows towards the origin on the \(\lambda=-1\) line and away on the other

1. (a) Stable node: eigenvalues \(-6\text{ and }-1\); eigenvector lines gradient −3/2 (\(\lambda=-6\)) and gradient 1 (\(\lambda=-1\)); arrows towards the origin on both lines
2. (b) Draw the line through the origin with direction \(\begin{pmatrix}2\\-3\end{pmatrix}\) (gradient −3/2) for \(\lambda=-6\): arrows point towards the origin. Draw the line through the origin with direction \(\begin{pmatrix}1\\1\end{pmatrix}\) (gradient 1) for \(\lambda=-1\): arrows point towards the origin. Other trajectories come in from far away alongside the \(\lambda=-6\) line and reach the origin tangent to the \(\lambda=-1\) line.
3. (c) The point lies on the eigenvector line of \(\lambda=-1\), so the solution stays on that line: \(x=4e^{-t},\ y=4e^{-t}\). Both amounts decay to 0 and their ratio stays constant.

1. (a) Saddle point: eigenvalues \(-4\text{ and }1\); eigenvector lines gradient −3/2 (\(\lambda=-4\)) and gradient 1 (\(\lambda=1\)); arrows towards the origin on the \(\lambda=-4\) line and away on the other
2. (b) It moves in towards the origin along the line, getting closer for ever but never reaching it (\(x=4e^{-4t},\ y=-6e^{-4t}\)).
3. (c) It moves away from the origin: for large t it follows the line of gradient 1 (\(x\approx1.4e^{t},\ y\approx1.4e^{t}\)).

1. (a) \(\det\mathbf M=2-3k\)
2. (b) \(k>\tfrac23\)
3. (c) \(\lambda=1\text{ and }2\): unstable node