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Unformatted text preview: 65.2400 TEST 3 December 1, 1995 Name Section Please answer all questions, showing your work in detail. You may use one 8.5 by 11 sheet of notes while taking the examination. No other reference
material, books, notes, or calculators are permitted. NAME _____________________________ SECTION ________________ __ 3. (20 points) The eigenvalues and eigenvectors of a certain matrix B are 1 1
T1 + 29E T2 lag Find the solution of the initial value problem in terms of real valued functions. NAME _______________________________ __
SECTION __ _____________ __ 4. (a) (10 points) Find the eigenvalues and eigenvectors of the matrix A = < 3 31 > (b) (8 points) The eigenvalues and eigenvectors of a certain matrix B are l —2
= _2 (1) =  = 3 (2) = ,
r1 3 E 7‘2 a g 1 Sketch a phase portrait of the system 23’ = Bar. (0) (6 points) Based on your phase portrait in part (b) sketch the graphs of 2:1 versus
t and x2 versus t for the solution of :r’ = Bar that passes through the point (1, 3). Label
which graph is which. XI ,1 (d) (6 points) The point 2: = 0 is the only equilibrium solution of :r’ = Bx.
Is it a node, spiral point, saddle point, center?
Is it asymptotically stable, stable, unstable? Underline your choice in each case. NAME _____________________________ SECTION __________________ __ 5. Consider the nonlinear system I It =$(2+y) =F(I,y)
y’=y(4—y—$)=G(Ivy) (a) (6 points) Find all equilibrium solutions (critical points), i.e., points where 10’ = 0
and y’ = 0. For one of these points both :1: > 0 and y > 0. Call this point P1. (b) (14 points) Find the linear system that approximates the given system near P1.
There are two ways to do this. One way is to form the matrix of partial derivatives F F . .
< G‘” G” > and evaluate 1t at P1. Another way Is to let x = 330 + U, y = yo + 11» Where
:3 1/ ($0,310) are the coordinates of P1, substitute for ac, y, 17’ and y’ in the given system, and
keep only terms that are linear in u and 21. Show that both methods lead to the same approximate linear system near P1. LAB TEST 3 65.2400 — 13, 14, 15 ' November 30, 1995 NAME ___________________ __._ SECTION _____________ _. 1. The function °° km: 1 10 km:
ux,t = "k2"”t/1°° ' —, =—/ 30—2 ' ——d
( ) gcke sm 10 ck 5 O ( I)SII1 10 :5 is the solution of a certain heat conduction problem.
(a) Write down the problem (PDE, BCs, IC) Whose solution is the given u(a:, t)
PDE:
BCS:
IC: (b) Find (to 5 decimal places) the numerical values of the ﬁrst two coefﬁcients
01 and C2.
01 = _____________ _.._ C2 = _ __________ _____ (c) Using ﬁve terms in the series, plot u versus at for t = 1. Sketch the graph below.
(k X (d) Estimate from the plot in part (c) the coordinates of the point Where the temper
ature is maximum. NAME ______________________________ __ SECTION ________________ __
2. Consider the system I’ = %1‘ + 2y, y’ = —3m — iy. (a) In the xy plane plot the trajectory through the point (—1, ‘2) for 0 g t g 8. Sketch the plot on‘jthe left hand axes below.
X ‘,.i_'_,_‘_ (b) Plot z versus t for the solution you found in part (a). The scene option in DEplot
is useful for doing this. Sketch the plot on the right hand axes above. (c) Estimate the ﬁrst three times at which :3 = 0 and y > 0. (We are looking for two
digit accuracy here.) (d) Classify the equilibrium solution at the origin as to type (saddle point, node, spiral
point, center) and state whether it is asymptotically stable, stable, or unstable. (e) Plot the given trajectory in three dimensional try space and sketch the graph
below. Again, use the scene option. ...
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This test prep was uploaded on 04/09/2008 for the course MATH 2400 taught by Professor Yoon during the Spring '04 term at Rensselaer Polytechnic Institute.
 Spring '04
 Yoon
 Differential Equations, Equations

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