solutionofexam_2MTHMTH205

# solutionofexam_2MTHMTH205 - Department of Mathematics and...

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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/QlEJCl-M5iokﬁ‘)+(678’ I L ‘ Ell-tangents 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.

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solutionofexam_2MTHMTH205 - Department of Mathematics and...

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