{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

527ex2sl

# 527ex2sl - 642:527 SOLUTIONS EXAM 2 x y and A = 43 12 FALL...

This preview shows pages 1–2. Sign up to view the full content.

642:527 SOLUTIONS: EXAM 2 FALL 2007 1. (a) Find the general solution of z = A z , where z = bracketleftbigg x y bracketrightbigg and A = bracketleftbigg 4 3 1 2 bracketrightbigg . (b) Give a careful drawing of the phase plane ( xy -plane) for this system, showing enough trajectories to indicate qualitatively the motion in each region of the plane, as well as any “special” (straight- line) trajectories. Your trajectories should be marked with arrowheads giving the direction in which the solution moves as t increases. Solution: (a) Since det( A λI ) = λ 2 6 λ + 5 = ( λ 5)( λ 1) the eigenvalues are λ 1 = 1, λ 2 = 5. The corresponding eigenvectors z ( i ) are found by solving ( A λ i ) z ( i ) = 0 : λ 1 = 1 : bracketleftbigg 3 3 1 1 bracketrightbigg bracketleftbigg x y bracketrightbigg = 0 = z (1) = bracketleftbigg 1 1 bracketrightbigg λ 2 = 5 : bracketleftbigg 1 3 1 3 bracketrightbigg bracketleftbigg x y bracketrightbigg = 0 = z (2) = bracketleftbigg 3 1 bracketrightbigg The general solution is z ( t ) = c 1 e t z (1) + c 2 e 5 t z (2) . This is an unstable node . (b) Here is a solution plot, from Maple. I don’t know how to get Maple to put arrow- heads on curves, but as is clear from the di- rection field or from the form of the solution, all trajectories are oriented away from the ori- gin. The straight line trajectories are parallel to the eigenvectors found in (a). The other trajectories are determined by the fact that as as t →∞ they are parallel to z (2) and as t →−∞ to z (1) . 2. (a) Consider the system x = 1 y, y = x 2 y. Determine its singular (equilibrium) points and classify each, insofar as possible, using linearization: for each equilibrium point, determine whether it is unstable, stable, or asymptotically stable, and if it is a focus, node, or saddle, if you have the information to do so. If one cannot determine the type from the linearization, say so.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 4

527ex2sl - 642:527 SOLUTIONS EXAM 2 x y and A = 43 12 FALL...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online