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Unformatted text preview: 65.2400  20, 21, 22 TEST 3 December 7, 1994 Name Section 20 21 22 Please answer all questions, showing your work in detail. You may refer to one sheet (8.5 by 11 inches) of notes. No other notes, books, references,
calculators, etc. are permitted. NAME _______________________________ __
SECTION 1. (25 pointS) (a) Find the eigenvalues and eigenvectors of the matrix A = < i :11 > , (b) A certain matrix B has the following eigenvalues and eigenvectors: 1 1 1 1
=—— (I): ' =___2' (2):
T1 27 E r2 2 17 E Find the solution of the initial value problem NAME ______________________________ __ SECTION __________________ 2. (20 points) (a) A certain matrix A has the following eigenvalues and eigenvectors: 2 2
11 1.§ 12 1 E ' < (i) In the arle plane sketch several trajectories of the system 23’ = Ax; be sure to include
the trajectory through the point (—4, —1).
(ii) Also sketch the graphs of 2:1 versus t and 132 versus t for the trajectory through (—4, —1). (iii) For this solution what is the approximate value of the ratio 1:2 / $1 as t ——> 00?
1? X1 qu1
l
l
' a t
l
i (b) A certain system 23’ 2 Bar: has the phase portrait shown in the diagram. What can
you say about the eigenvalues and eigenvectors of B? NAME _______________________________ __ SECTION ____________________ __
3. (25 points) (a) Transform the equation
4y” +— yy’ — 200sy = 0 into a system of two ﬁrst order equations. (b) Consider the system 20’ = CI: — 3:312, y’ = 4y + 1:212.
(i) Find all critical points (equilibrium solutions).
(ii) Consider one of the critical points that is different from the origin. By examining the
appropriate approximate linear system, classify this point as to type (saddle point, node,
spiral point, center) and state whether it is asymptotically stable, stable, or unstable. LABORATORY TEST 3
652400202132 December 9. 1994 NAME _____________________________ __
SECTION _________ _ 1. Consider the system
dm/dt = «31: + 01y, dy/dt = 91' — 2y. (a) Let a : 2. Draw a direction ﬁeld for this system on a square centered at the
origin. On the axes below sketch a few trajectories of the system. Also classify the origin
as to type (node, spiral point, etc.) and state whether it is asymptotically stable, stable. or unstable. (b) Now let a = 5. Repeat the steps from part (a) in this case.
‘1 (c) Find (or estimate as best you can) the critical value of a where the behavior shown
in part (a) changes to that shown in part (b). Give your value for amt and also describe brieﬂy (but clearly) what you did to ﬁnd it. acrit = — 2. Consider the system = —(1/4)x+y — $2, 2—: = ——2$+:ry. ﬂ
dt Draw a direction ﬁeld for this system on the rectangle 4 S x S 4, —4 S y S 4. It may also be helpful to include a few trajectories in your plot in order to answer the following questions. (a) From your plot identify all critical points (equilibrium solutions) and list them below. (b) For each critical point classify it as to type (node, spiral point, etc.) and state whether it is asymptotically stable, stable, or unstable. (c) One critical point is asymptotically stable. Sketch the basin of attraction for this critical point. A j ...
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 Spring '04
 Yoon
 Differential Equations, Equations

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