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Midterm2_key - MATHEMATICS 201 SECTION A01 NHDTERle...

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Unformatted text preview: MATHEMATICS 201, SECTION A01 NHDTERle Jubr6,2011 Name: Student Number: Duration: 50 min. Maximum score: 30. The test has 5 problems and 5 pages. '- Full answers, please. Justify your statements. The Sharp EL—510R scientiﬁc calculator is the only calculator allowed. No other aids such-as books, notes or formula sheets are permitted. 1. Find a fundamental set of solutions to the differential equation 3/" — Zay’ + (a2 + b2)y = 0. ; 2. MLIl-GM—yaelo ;»3 r—“f/ 2, ¢ ‘ Clix -' I) 5, ISéZ/‘le—a :5 £00?!) [at a a sick“) are {a s 7, l 7 «ML, {lv‘x/‘Q - l/JL ’59 Slow {Lo} me “A” l of, tax ‘ Ml l) Q” g” bk l ' Q '0 ‘s\. “B: ‘3. , .. 1x _ Ml?) ’Vl l S ’ ‘b( g smut c.) M . ta- bk) ea (“v “WW” ’ 1 “ 16x1 « 1/“);ng bk) , §AEK (“7(3' ‘ e “chit” ’ .» 2,6le . ‘ ‘1 r g. l; «LBNV‘3 1; b L; »% U Midterm 2, July 2011 MATH 201 Page 2 2. (a) Write the following system of ﬁrst order differential equations as a second order d.e. for a(a:) 0’] r '71 t (1'66) = 3b(a:) + 3 sin(3a:) ] L. b/(il?) = —3a(:v) / / _ r (RI/(x) »-3IO(4~) = 9663 (SR) O‘\ﬁ(*r~) if 0~ 3 '9 C35 (3%) (b) Solve the obtained non—homogeneous differential equation. 7. Jr ,_ \V ;L- l M +97»; 3) mum ma . (F's. \ W w T; r 5" U) F 01 4:) ' g“? I“. "=3 Sc r .7 _ (km; : (1,“; a Cx i7.) 73/; &>L a.) r C CD '5‘“ 1» Ci got 3‘“ L P ‘ gt, Midterm 2, July 2011 _ 3. MATH 201 (a) Find the general solution of the homogeneous equation 4 my" — 3y’ + Ey = 0. Hint: You can rewrite this equation as a Cauchy—Euler equation. Page 3 . fl : i \9 ~ "k L f ._ 4‘ 1‘ g G M f‘ " " (L t S \x 1’ L F “b 13 i ‘i “l is 7., 0% xi! 5’ ' C, , t m r \ M . a .r ., ‘ : ' s M w z.) m, (mlm'\l”3‘m *ZC) Nb ﬂ .2. v — M( m.1) '2. S» 'M ,AMJQI‘ '50 we») (M~l)§0 A 97 a z- ' 3’V\ \$ L K" CD7 s c l a» C 7" I“ 7” . - l L (b) Find the general solution to the following differential equation I 2. . N 4 II__ / 4 __ 3 I '6 L { ,— s L my 3317+ my —— as ‘3 .1" W2, — u i U‘ “<3 L 907— 1M“ *1 I F- ' m1 4 7L 2 l . L 2',“ 23A 0 k {Mk 7” ﬁx «a LTJ‘“""\ 1‘ 1x 1 . , w. : __ 1 V‘ A ' 1 n W n i ./ s ______i/ T l l / W \$5 2, '7‘:L i 51;“) pa / / 61‘,\ g .. (7V M 1‘ z \ 7"" r L in K g 1 r , 3‘12, \ 4 N2! [53' \ I Z I ,K . U‘k ( bull \‘Zw T" 5 OK, 7.3 UKL 7 /Z '1. M g 7 L! a 1/"); [‘g’zi/ﬂ- i ‘ ’7'» IA 4— m .- 1 “K (A K 9..” - k / (7 ‘5 a N 2: \E) y c x J. CL r 4 ,/ L \ (V, i3 l a 2 » i 1,3 I 4‘ f l 7k. f R + X / Midterm 2, July 2011 MATH 201 Page 4 4. A body With mass of 0.5 kg is attached to the end of a spring which stretches 2 m with a force of 8 N. The surrounding medium imposes a damping force equal to twice the velocity. (a) Find k, the spring constant. 141‘ ii: 1—”- i» , :> s lql) "*3 i V, I.- kiss (b) When the mass reaches equilibrium and has no instantaneous velocity, its sup— port begins to oscillate vertically according to f (t) = 20 cos(t). Determine the differential equation of motion With its appropriate initial conditions. W] I: / r _ .. . 22> 1%,: 2% +4 7L: 1‘3 C” d) L \ "XLORG w/[olsd (c) The equation of motion associated with this system is + @ cos(t) + j—Zsin(t) 56 72 ) 13 13 — —— cos(2t) — 1—3 sin(2t) x(t) = 6—2t( 13 Identify the transient term (solution) and the steady—state term (solution). ,- l , r i A - (1 ll‘eAAS-‘ai ﬁnt’M #3 Q, ( A: f if (3) i"? Ii "‘7 06 Jim/i ‘3 254 l '3 ‘i —e SV\( 1 I (.3 t3 Mtg/ﬂ it 5L; m 0%; “IE7W’F‘M (d) What is the amplitude f oscillaiions after a very long time it J “EMT, 5;} 3le 442 I A I?) l?) .i . 5‘ i3» ’ Aka .9611 Midterm 2, July 2011 V MATH 201 Page 6 [5] 5. Find a family of solutions for the given differential equation. Also, ﬁnd a singular solution. No marks will be given for guessing the singular solution; you must provide a solid piece of evidence that leads you directly to conclude that it is a singular solution. 1 (y ~_ 2)y// _ §(yl)2 : 0 dis/2 0!?“ C/h Ci 2.. I )' (a—l)b\m’-\€v\ ,6 1..) L&(ula-7c)u_’\iu)r€l it 5 I 5 C w 5’ 9 :3 ‘3 :0 5 \6 :5 as j 01w i CW3 W} 43 ;> \n Ml — ii M 13' xvi—>6 ‘3 in Ux «lira (all) "k C, L M s L M (w (31/ l ‘ C ‘3 ::;f' 1 ‘9 in ex .. in nix-6,7,) : C ’5 ‘a_z \; 34L ' C 2 ,5 a ,L r C 7‘» T 2 Fwd yd UK 3—”) Z L; all 5) l a I i) l . Cr- rd/ (1 4 (:11. -r * 2‘ ,3 J ,1 = E), 1 + ,J 0’ ‘3' l g, i) L 2— JEONW‘L ) S M W \asl H C~ S‘Mylw éomﬁ‘“ 5%; 5593qu 451i»: (if-e f 5' 3 I?» Jaw {225' END 0“?) “i1: 95ft A Fixw AT 82L} E/ < 1 j;— ' . a J“\" 8%; CKS}VM{)IP~ *t’J (irr’w— i 3L ‘6“ 2 n a ...
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