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Exam2Solutions

# Exam2Solutions - M427K Exam 2 Question 1 1[3 Points Find...

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Unformatted text preview: M427K Exam 2 March 22, 2007 Question 1 1. [3 Points] Find the general solution of the homogeneous second order constant coefﬁcient ordinary differential equation y”(x) + 21/(03) - 821(96) = 0 Choroderisvh‘c equoﬁm I‘s m1+2m —‘8 =0 when Poclor 12123 [D (m+‘t)(m~7~3 =0. THUS m=2 or mr-Ll. 7. v—4x. lj:Q_‘Q,I+C1€ 2. [3 Points] Find the general solution of the homogeneous second order constant coefficient ordinary differential equation 9y"(\$) ~ (ii/(w) + 3/010) = 0 Chqmderishc equation (3 qm1—6m+[=c>, 0T m1“2/3m‘+‘/q=0 .This' IS (m~%)1=o 3. [4 Points] Find the general solution of the homogeneous second order constant coefficient ordinary differential equation Wm) ~ 41/(96) + 8y(w) = 0 Chomderishc ecluodion ts owl—4m +\$§=O This is \$410244 =0 so the roofs are m=lt1i H2an ij = Q, e:Z>< cos (27c) Jrgellsm M427K Exam 2 March 22, 2007 Question 2 l. [5 Points] Find the general solution to the inhomogeneous second order constant coefﬁcient ordinary differential equation y”(t) ~ 4y’(t) + 13y(t) = 40 sin(3t). Chomcierislic QCL“ ml‘flm + )3 =<m ‘lelq =0- 30 BAﬂC'l'IOH lg \ﬁef- 0162 C03 +961ng C’MQ331 ‘ Lj’f Asm at “t B COSZ‘t no overqu. 3?‘ = 73A cosy: -%B 3m3+ 3P“ =‘q/5rsm3l: cc] Ecog3t' Substiluie (31A H15 + BA) 3m 3t +( 4115 41/4 + 13133 COS gt =40sm’5t (4A +ms )smgt + (—m +453 Cos at = 40sm3'l; 2. [5 points] Find the general solution to the inhomogeneous second order constant coefficient ordinary differential equation y”(t) + y’(t) — 2y(t) = 4e‘ + 5e2t Chomdeﬁs’li c cg" ml+m —2 : (mi-1) (rm—D =0 ~ ‘ v ~21: +t So comylcmm‘l'cinj limchon IS ch=C‘ e, ‘t (116 , Guess (A?) :AeJC . Overlays 3o molhplg 3F, 2 Atct no ovaqu. 9P: 2 Ate:t +Ae’C «a? = Atet +2Aejc' Subsl‘l'iuie . ‘3)“: el’ =— Lick =§ A=+/3 Quqsﬁom "U cjrok 4A +11 B =40 Equah‘ng wemcienh 'v-Im + 4r rs "=0 lmmgdwojrelj we 522 i‘ho’r B=\$A cmd H’me 4OA=40 so A"! and 8:3. The award Soluﬁon {3 HWQPQPDFE L3: Ema: Jr 3C033t “L 96123380 +C1€11f8m83f> QUQS’IVOH 2.1 dd. Gums Lj = A 6% no OVQHQP L“; = 1A6“ g}: = 4A a?“ gubﬁﬂujre 4A alt * 56% A = 51+. so (3?; = 5/461JC’ Thus Hm RAH 39mm "balk/dim is 4 ‘ 1» at. E a: /3Jg€J°+5/4et+cle +C2e, M427K Exam 2 March 22, 2007 Question 3 [10 Points] Find the general solution to the second order linear ordinary differential equation 3:23;” + Bacy’ + y = 0 given that y1(m) = i is a solution. L9} szvbchﬂpcl z v90 31—. was; - v00; 5“ Wm; — vbo 2/12 ﬂue»; Suhsli‘iuia CqH we lawns should camel), {‘(vl'bclslz — v'wiil +‘%7c(v‘(xl VD =0. xv“(x) + VIOO =0 . lSl order [mew d ‘ (El xv‘bcﬂ ‘ 0 I V‘bO = C- vlbO ~— C/x. vbc) =Cln7c+ 9. Thus ’th general soluiiom 1‘3. 9(x)=vbc)cj.(x) : cit/ix + D VI. M427K Exam 2 . March 22, 2007 Question 4 [10 points] Find the general solution to the inhomogeneous second order linear ordinary differ— ential equation t2y”(t) — 3ty’(t) + 3y(t) = —2t2 + 6 given that y1 = t and y2 = t3 are solutions of the associated homogeneous ordinary differential equation fizz/’03) — 3733/“) + 3W) = 0 Warning: This equation is not in standard form. mega equaliom ‘ I / V\ U. £3, + Lag; =0. Assummq Q QCL of 3.5+ £4,132” =9C+\. is m Sidnde Perm. For our equqhom that; ore. aft + u; J? :0 o {21(u,’ + uz’BJcll = “ZJC7‘+6 Q) Ex owing and multiplying (D t. P aft1 +u£tq= b3 @xl: (1/13 + a; 3% = {the (7;. u; R" e. —ltl+6.3 @ “CD xt‘ _; + = = n- A} integral??? we oblom.. ‘ ‘ ulz£+gt+cl U2: Vt“ /t3‘+CL Remembef g=umg1+ M191. 3 (j: to, + tguz: {1493+ Q‘i: + {l‘l Jrth = th+1 +0,{ “LP. 5 M427K Exam 2 March 22, 2007 Question 5 [10 Points] Find a series solutions about x0 = 0 to the second order ordinary differential equation y”—wy’—y=0 Find the recurrence relation and the ﬁrst three terms in each of the two linearly independent solutions. Lcl «A = : armor? n—O 00 n-I m V‘ 3=20nn7€ so xg‘iarmx' n=n Rcmdcxmg with m=rr1 we gel UllzgoelmCm‘thWH—llxm' Subslhlng m w n E Cln+acn*'1)(m'l) Xn ‘ Z Gin nxn -' 2 an X :0 ":0 "zl h=o Extradv‘ng lama) and COVDOlldCthHQ So 3 : QDC I aim" +§x4+~~>+ql><+ 3L>< *{%X§*“ﬁ>. ...
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Exam2Solutions - M427K Exam 2 Question 1 1[3 Points Find...

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