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Unformatted text preview: Department of Mathematics and Statistics
American University of Sharjah Second Midterm Exam  Spring 2010
MTH 205  Differential Equation Date: April 14, 2010 Time: 5:00—6:15 pm Instructork Name: C. Alobaidi A. Sayfy Z. Abdulhadi
Q '\ Student Name: Xx» Student ID Number: Time of the Lecture: M I . Do not open this exam until you are told to begin. 2. N0 Questions are allowed during the examination.
3. This exam has 5 questions. 4. Do not separate the pages of the exam. 5. Scientific calculator are allowed but cannot be shared.
Graphing Calculators are not allowed. 6. Turn oﬂ all cell phones and remove all headphones.
7. Take oﬂ your cap. 8. No communication cfany kind. Student signature: ’
WM“ Q]. (35 points) Multiple Choices: Circle the correct answer. I. The existence and uniqueness theorem guarantees a unique solution to the IVP: xzy'+ x y’+—I—y=~/;, y(l)=2,y’(l)=l on:
5¢x x——3
(A) (‘5, 5) f (B) [0. 3] (C) (5, °°)
\ (D), (0. 3) (E) (0, 5)
\fV II. All the following pairs of functions are linearly independent on (~00, 00)
’ L/ exept:
(A) 1, e“ (B) sinx, cosx (C) x, x2 e", eN3 (E) x, xe‘
1 III. The period of the motion described by the equation a x’y + 18x = 0 is:
(A) 3 35 (C) 5 (D) 6 (E) 1
2r 3 6 IV. The amplitude of the motion which is describe by the WP
2y'+8y = 0, y(0) = 0, y’(0) = 6 is: (A) 2 @3 (C) 4 (D) 6 (E) 8 V. The motion which is describe by the equation x'+ 2x’+ x = 0 is: (A) simple harmonic (B) overdamped (C) underdamped critically damped (E) beats VI. A fundamental set of solutions of y“) +4ym +4y"=0is:
(A) 0, x, (2‘, e" (B) l, x, e'z" (C) 1, x, x2, e4" @ 1’ x’ 84'" “'2‘! (E) 1, 9“ VII. The form of a particular solution of ym+ '= 2 +003 2x is:
(A) A + Bcost (B) A + Bsin x + Ccosx @ Ax + Bsin 2x + Ccost (D) Ar + Bxsin 2x + Crcos 2x (E) A): + 88in x «t» Coos x Q2. (15 points)
Given that y, 2 e“ is a solution of the differential equation: xy'—(x+ l)y'+y =0, x>0.
Use the reduction of order technique to ﬁnd the general solution . bum ikawL “1‘2” 3 O ) 7C 7< ' i w '7<‘ %\ 1 87”" (AX Q3. (15 points)
Use the method of undetermined coefﬁcients to ﬁnd the solution of the I.V.P. 2y'+4y'+2y=e‘+x , y(0)=1, y’(0)=0. <3 \NXW ‘WW i»; ,3
__ “W3 ,><~._.,2:,
QC€7<% QALV LAC 6% +'ZA7L*1K+2C€ “C h §C%MC*ZCZ )C\
8C:\~:‘>C‘:;] A) l:~28 , WMNK Q4. (15 points)
Use variation of parameter to ﬁnd the general solution of the differential equation: xzf 2y = x2 ./ ,. mg.“ / , .4; .—~. 7/ C) WW Om”2> W\?N Z
[ww+\> :cg
m "if, “ W _ C1 7;) VY’\ ,\ 74 + v ~ ’ l 75 01 >4 : L3 LMX l “' \ , 3 m 7< 2_ “‘7 M "1 * I, .
\ O X ()1 x 71 M 6’} 7“”? 05. (20 points)
A mass weighing 128 pounds stretches a spring 2 feet. The mass is initially released
from a point 6 inches above the equilibrium position with no initial velocity and by
simultaneously applying to the mass an external force F(t) ==85in(4t). Assuming no
air resistance: 21) Find the initial value problem corresponding to motion of the spring. b) Solve the [VP in part (a) to ﬁnd the motion of the mass. c) Find the position of the mass at t: It/ 43nd state whether it is below or above the equilibrium position.
b0: W‘VCB 2) 3‘1 §‘1W\i31):>m39
e 3 \
Ribti g$fmbﬁf Li 7< ” 4i 6L} >< .., <3 u t , 7m» : .. p
mix 7i H l b : 1 jglﬂ Ll ‘QON’CQCQ U” W a a)» W“)???
\U hag/WE,“ ‘ EEK; {Ni ix 1 C\ (as; AV 3‘ <1, 3\NV’\W% ] l 74]”)12i/QlEJClM5iokﬁ‘)+(678’ I L ‘ Elltangents emu, 7<FltuLlﬂiSivxgarKAm A(QM A i Cagtﬂt] + gen/m; . //; 53t~lbiwﬂq ii ...
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This note was uploaded on 02/12/2012 for the course MTH 205 taught by Professor Sadik during the Spring '11 term at American Dubai.
 Spring '11
 sadik
 Differential Equations, Equations

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