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Unformatted text preview: K LABORATORY TEST 2
65.2400—04,05,06 October ‘26, 1993 NAME ___________________________ __
SECTION ________ __ 1. The differential equation (u’(t))3
3 u"(t) — u’(t) + + u(t) = 0 is known as Rayleigh’s equation. It describes a certain type of oscillating system, and is
somewhat analogous to the pendulum equation of Assignment 7, except that the nonlinear term is different. (a) Choose a set of initial conditions, that is, values for u(0) and u'(0). NOTE. Do NOT choose both 11(0) and u'(0) to be zero, and for practical reasons, do
not choose them to be too large; numbers in the range from —5 to 5 are recommended. Use DEplot to plot the graph of u versus t for the interval 0 S t S 25. Sketch the graph in the space below. Be sure to put a scale on the u axis.
4,; A _.__L..__.M,._____..___.L.__..__. M ._.. _ l wean“ _.. ...l._.___.._.i_..1, t
5 [0 I5 2 o 7.5 (b) After at most a few cycles your plot in part (a) should show a steady periodic oscillation. Estimate the amplitude and period of this oscillation. Amplitude _______ _ Period _______ __ 2. The displacement u(:::) of a certain elastic beam under a transverse load satisﬁes the differential equation
d4u ——— = 5x cos 2:.
(1.734 Suppose that, in addition, u(a:) satisﬁes the four boundary conditions
u(0) = 0, u'(0) = 0, 11(1) = 0, u"(1) = 0. Note that two of these conditions are at x = 0 and the other two at x = 1. (a) Use dsolve to ﬁnd the solution of this problem
NOTE: Recall that u"(1) is entered into Maple as ([email protected]@2)(u)(1). Plot u(x) for 0 S a: S 1 in the space below. Be sure to put a scale on the u axis. “t ' 0.1 0.1 0,1. 0.8 (.0 (b) Estimate from your graph the x coordinate of the point where is maximum, that is, estimate the point of maximum displacement. IE: __________ __ (c) Estimate the maximum value of u(x), that is, the value of u at the point you found in part 65.2400  04, 05, 06 TEST 2 October 29, 1993
Name Section 04 05 06 Please answer all questions, showing your work in detail. One (twosided) sheet of notes is allowed, but no other books, notes, calculators, or other
references. Problems 3 and 5 are worth 20 points, while Problem 4 is worth 30 points. 65‘240004,05,06 NAME ___________________________ __ 3(a) The eigenvalues and eigenvectors of a matrix A are _ . (1)_ ~1+2i I __ __ . (2)_ —1—2i
rl3+2z3 é —'( >9 7.2—3 2c, 6 — . Write down the general solution of the system x’ : Ax in terms of realvalued functions. (b) The eigenvalues and eigenvectors of a matrix B are 9 _
1‘1 : —1, 5(1) : 7‘2 :2, 6(2) : < . Sketch the phase portrait of the system x’ = Bx. A
V X 2. r. 65240004,05,06 _ NAME ____________________________ _ 4.(a.) Find the solution of the initial value problrm 2y" +2y' + 5y ; 0, y(()) = 1, y'(0) = —1. b Find the eneral solution of y" + 33/ + 2y = 3J;+ 26".
g _ 65.240004,05,06 NAME ____________________________ __ 5(a) A trajectory of a certain differential equation in the try plane is shown below for
0 S t S 15. On the other axes plot at versus 1‘ and y versus t. Be sure to put scales on your axes and to label which graph is which. (b) The Wronskian of the functions and g(2:) is 2 2x3. If = x2, ﬁnd Your Name 65.240 November 5, 1993 Circle your Section Number: 10 11 12 20 21
(2 pm) (3 pm) (4 pm) (9 am) (10 am) Please do all 4 problems, showing your work clearly and
in reasonable detail. Give reasons for your conclusions. One 8.5 by 11 inch sheet of notes may be used. Other
materials (books, papers, calculators) are not permitted
during the exam. Problems 1 and 2 count 25 points each, and Problems 3 and
4 counts 15 points each. The Lab question counts 20 points. 1.(a) (12 pts) Find the eigenvalues and eigenvectors of the matn'x A II
A
F‘H
I
y—ah
v (b) (i) (7 pts) Find the realvalued general solution of the equation y"  2y' + 5y = 0. (ii) (6 pts) Find the solution that satisﬁes y(0) =0, y'(0) = —2. 2(a) Suppose y : y(x) satisﬁes the equation 9y" + 6y’+y = x. (*) (i) (12 pts) Find the general solution of (*) . (ii) ( 3 pts) On the next page are four graphs, labeled 1 through 4. Write the number of
the graph that shows a possible solution of (*). (b) (10 pts) Find the general solution y = y(x) of y" + 9y = sin(3x) . HINT: Use as many shortcuts as you want. 13 3. (a) ( 9 pts) Below are three equations. On “he next page are six graphs of solutions u(t),
labeled 1 through 6, each satisfying u(O) = 0, u'(0) = 1. In the space by each
equation write the number of the graph that shows the corresponding solution. u" + % u‘ + u = sin(t/2)
u" + 31— u' + u = sin(t)
u" + % u' + u = sin(2t) NOTE: The scale on the taxis is the same for all six ﬁgures, but the scales on the
uaxis are different. (b) (6 pts) The differential equation
x2y" — 4xy' + 4y = x3ex (*) has two solutions y1(x) = x and y2(x) = x4 for the corresponding homo geneous equation. Set up an expression for the general solution y(x) of (at). NOTE: You need NOT evaluate any integrals that appear in your solution. (a) (b) (1) (2) (3) (4) (5) 9 ‘ : Consider systems of the type 1“ = Ax, where A is a 2 x 2 constant matrix and = [:8 . about the system or its solutions. For each of the 5 cases below, the ﬁrst column gives some information (5 pts) In the second column, put U if the critical point x = 0 is unstable, S if it is
stable, and AS if it is asymptotically stable. (10 pts) In the third column, sketch a possible phase portrait, showing just a few curves
near x = O. (b) Given Information General solution: 3
_ 4t —1 —t
x(t) — cle l: 3:+c2e Ll] 65.240 LAB EXAM 2 Name
Sec. 20 21 10 ll 12 “W Books, notes, Maple, etc. may be used. No discussion with other students. ““ A system of differential equations containing the parameter a is given by
dx
aT—3x2y, dt=ax—y.
1. Enter the equations in a way suitable for use in dsolve. (a) Suppose a = 0. Use dsolve to ﬁnd the general solution. x(t) = _____'_________, y(t) = (b) What kind of equilibrium solution (critical point) is the origin?
(c) Find the solution that satisﬁes x(0) = 1, y(0) = 1. Use Maple to plot x versus t
and y versus t, and sketch your solutions on the axes below (left). 2. (a) Suppose a = 2. Use dsolve to ﬁnd the general solution.
X(t) = ______.____. y(t) =
(b) What kind of equilibrium solution is the origin? (c) Find the solution that satisﬁes x(0) = —1, y(0) = 1. Use Maple to plot x versus t
and y versus t, and sketch your solutions on the axes below (right). 3. As a changes between the values in parts 1 and 2, the type of equilibrium point changes. (a) At what value of a does the change occur? a =
(b) Find the general solution for this value of a. x(t) = ______________, y(t) = Part lc Part 2c FOR EXTRA CREDIT: Use Maple to produce a phase portrait for the a value in part 3. Sketch the
picture on the back of the exam page. ...
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 Spring '04
 Yoon
 Differential Equations, Equations, 3 pts, 5 pts, 6 pts, Stability theory, 3j

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