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Unformatted text preview: LABORATORY TEST I3 65.2400 ~ 07. 08. 09 April 25. 1994 NAME ____.____~____._______ SECTION ma
1.Consider the system I’ = I + 2y. 3/ = (1/2)3: —— 3y. (a) Draw a direction ﬁeld on the square —2 g r 5 2, —2 g y g 2.
From an inspection of the direction ﬁeld classify the origin as to type saddle point node spiral point center and whether it is
asymptotically stable stable unstable (b) Plot the solution that passes through the point (1. 2) and sketch its graph on the
axes below. “3 (C) For the solution in part (b) sketch the graphs of I versus t and y versus t. for
O S t < 5. on the axes below. Put scales on the axes and label the graphs clearly. _ L3 (d) For large t what is the approximate ratio of I to y for this solution? ratio = ‘2. Consider the system 13’: 9. y’ = I  (ll/(3)1J —1‘1/4‘y. (a) Draw a direction ﬁeld on the square —1 _<_ r g 4. —4 S y S «1. By looking at
the direction field determine the approximate location of each critical point (equilibrium
solution). (b) Classify each critical point as to type (saddle point. node. ...) and state whether
it is asymptotically stable, stable. or unstable. 'n (c) Plot the trajectories through the points (0.2) and (0.3) for 0 S t S 20. Use a
stepsize of 0.1 to obtain smooth curves. Sketch the trajectories on the axes below. ‘1 (d) There is a trajectory between the ones in part (c) that behaves in a different way
from both of them. What happens to this trajectory as t —~ 00? 65.2400  O7. 08. 09 TEST 3 April 27. 1994 Name
Section 07 08 09
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. Problems 3 and 4 count 20 points each; problem 5 counts 30 points. "' "MW NAME _______.__.___.___..____ SECTION _____..___.____ 3 (a) Consider the partial differential equation
uxz+uyy+4u=0 Assume that u(:I:, y) = X (I)Y(y) and find ordinary differential equations satisﬁed by
X(:c) and Y(y). _:
l (b) A system of equations 93’ = Act has the phase portrait shown below. What can be said
about the eigenvalues and eigenvectors of A? NAME _...._______ ______ __. SECTION ______.. _____ _ 4. The eigenvalues and ,eigenvectors of a certain matrix A are 1
=—1+2', (I): 
n z E 3+42‘ ’ 1
=—1 ', (2) = .
T2 21 5 (3—4,) (a) Find the general solution of x’ = AI in terms of realvalued functions. (b) Also ﬁnd the solution that satisﬁes the initial conditions NAME .__._______________ SECTION _____________________ 5. Consider the system I, = y, y’ = I  (1/6)133 — (1/4)?! (a) Find all critical points (equilibrium solutions). (b) Find the linear system that approximates the given system near the origin. Based on
the linear system determine whether the origin is asymptotically stable, stable, or unstable. Also classify it as to type (saddle point, node,...). CONTINUED NEXT PAGE... (C) Choose one of the critical points that is different from the origin. Find the approximate
linear system valid near this critical point. Based on the linear system determine whether this critical point is asymptotically stable. stable. or unstable. Also classify
it as to type (saddle point. node....)_ (d) One of the trajectories of this system is shown below. 0'1; the other set of axes sketch
the graph of 1‘ versus t for this trajectory. ‘1 ...
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 Spring '04
 Yoon
 Differential Equations, Equations

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