Lab 6 -- Eigenvalues and y'' = -by' - cy
Lab 6 (click here for pdf) concerns solving the system y'' = -by' - cy.
- To express the system y'' = -by' - cy as a X' = AX (where A is a matrix and X, X' are vectors), let y' = v. Then y'' = v' = -by - cy. Then let X = [y, v] and let X' = [y', v']. Now find a matrix A such that X' = AX. Now, evaluate det(A- lambda *I).
- Use the command fplot('x^2/4',[-4,4,-4,4]). x-coordinates on this plot represent b-coordinates and y-coordinates on this plot represent c-coordinates.
a.) As the handout says, eigenvalues are imaginary for (b^2)/4 < c. Label the region satisfying this constraint on your plot as "NR."
b.) Do the quadrant analysis as per lab instructions to find regions PP (where both eigenvectors are positive), NP (where one is positive, and one is negative), and NN (where both eigenvectors are negative). Label these regions.
- Follow these lab instructions closely. Be careful not to forget to draw your sketch of the small phase plane portrait.
- When following these instructions, remember to enter the A such that X' = AX that describes your system, using the values of b and c from problem 3. How to find the general solution for X' = AX is described pretty well in section 5.3 of your text. See especially pp. 367-368.
- Use the behavior of your solution for y as t approaches infinity to justify your responses to 3(a).
- Repeat exercises 3 and 4 for a (b, c) in NN.
- Repeat exercises 3 and 4 for a (b, c) in NP.
- You might want to see p. 379 in section 5.4A of your textbook to help you answer this.
Conclusion: Be sure to describe the nature of the eigenvalues (are they real or non-real?) and the phase plane portrait for the following conditions:
1. b^2 - 4c >= 0 and ...
b > 0, c > 0
b < 0, c > 0
c < 0
2. b^2 - 4c < 0 (which implies that c > 0) and ...
b > 0
b < 0
b = 0
That should be it for this lab.