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Unformatted text preview: Name: ________._.____.___———— MATH 2243  SPRING 2005 Student Number:
FINAL EXAM, Form: TA: ____________—————— Date: __________________ There are 200 points and 13 problems on this exam. The number of points for each
problem is indicated. The ﬁrst 6 problems are multiple choice, and you should circle the
correct answer. There will be no partial credit for the multiple choice problems. Problems
713 are free response. For these problems, please do your work in the space provided and
show all work. Partial credit will be given. However a correct answer may not receive full
credit if the justiﬁcation is incomplete or incorrect If you need extra space, work on the back of the pages. Please clearly label all work. a " ‘ " “ " ' ‘ '  l score Possible 16: Multiple choice 1. (10 pts.)
Which one of the following sets of vectors is linearly dependent? (a) (1,1,0,0,0),(0,0,1,1,0) (b) (1, 1, 1), (0, 1,0), (0,0,2) (c) (1,1,—1),(—1,1,0),(0,2,—1) (d) (1,0,0,0), (0,0, 1,0), (0,0,0, —2) (9) none of the above are linearly dependent 2. (10 pts.) An integrating factor for the equation tzy’ + 1533/ = 5 is (a) 217
(b) e”
(C) 6‘
(d) et2/2
(6) 6154/4 3. 10 ts. The equation y” — 2y’ — 3y = 56‘ — 4156“ + 2sin 3t has a particular solution
p
of the form
(a) clet + Cgeat + 03 sin 3t
b) clef + (Czt + c3)e3t + c4 sin 3t
((1) clet + t(czt + 03)e3t + 64 cos 3t + c5 sin 3t
(d) cle” + (Czt + 03)e3” + C4 cos 3t + c5 sin 3t
( ) cltet + (Cgt + c3)e3t + c4 cos 3t + c5 sin 3t A e 4. (10 pts.) Which differential equation is separable:
d
(a) 5% = (sin (w2>)<cosy)exy
(b) (22: + y2)d:r + sin azdy = 0
dy _ sin (xy)
(C) dm — x — 2 (d) 2—: = (COS 3/)(sin (562))em‘y (9) none of the above 5. (10 pts.) The dimension of the vector space spanned by the columns of the matrix 12 0 7
1 3 —1 9
11 1 5
1 5 —3 13 isequalto (a) 2 (b) 3 (C) 1 (d) 4 (e) 0 6. (10 pts.)
Consider the system of equations $131 + I2 = 2
£131 — 21132 + $3 : 1
3x1 + axg = 5 involving a real parameter a. This system has (a)
(b) a unique solution for all real values of the parameter a a unique solution for all real values of the parameter (1 except one value, and
inﬁnitely many solutions for one value of a a unique solution for all real values of the parameter a except one value, and is
inconsistent for one value of a a unique solution for all real values of the parameter a except two values, is
inconsistent for one value of a, and has inﬁnitely many solutions for one value
of a a unique solution for inﬁnitely many values of the parameter a, has inﬁnitely
many solutions for inﬁnitely many values of a, and is inconsistent for inﬁnitely
many values of a 7. (20 pts.) A cold beer with a initial temperature of 35°F warms up to 40°F in 10
minutes wile sitting in a room with temperature 70°F. According to Newton’s Law of
Cooling the temperature T(t) of the beer at time t satisﬁes the differential equation g
dt with initial condition T(0) = 35. What will be the temperature of the beer after 20
minutes? = k(70 — T) 8. (20 pts.) Solve the initial value problem 2/” + 10y’ + 9y = 0, W) = 2, y’(0) = —3 9. (20 pts.) Find a particular solution of
y”+3y'+2y =sinet (1) Hint: Use the method of variation of parameters and the fact that f eztsin etdt =
sin 6‘ —— 6t cos 6‘. 10 10. (20 pts.) Find the eigenvalues and eigenvectors of the matrix A: OJNb—t
[Or—IO
l
[O 11 11. (20 pts.) Show that the origin is a stable equilibrium solution for the Liénard equation
z'r'+(a:2+1):t+w=0 Hint: Transform the equation into a ﬁrst order system and use linearization. 12 12. (20 pts.) Find the general solution to the equation x’ = Ax, where A = i: _11 i ] . 13 13. Consider the nonlinear system x' = y+x\/:r2+y2 (3)
y’ = —:r+y\/$2+y2 (a) (5 pts.) Show that the critical point (0,0) is a (stable) center of the corresponding linear system. Hint: the Jacobian matrix of (3) at (0,0) is [ _01 3 (b) (10 pts.) Using polar coordinates we can write system (3) as
T! = 72
(9’ = ~1. Solve this system to show that the critical point (0, 0) is a (unstable) spiral point
of (c) (5 pts.) Explain why the results of (a) and (b) are not in contradiction with the
theory of linearization described in Section 7.2 of the textbook. l4 ...
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 Fall '08
 BOBKOV
 pts, Czt

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