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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 mrLl. 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 $410244 =0 so the roofs are
m=lt1i H2an ij = Q, e:Z>< cos (27c) Jrgellsm M427K Exam 2 March 22, 2007 Question 2 l. [5 Points] Find the general solution to the inhomogeneous second order constant
coefﬁcient ordinary differential equation y”(t) ~ 4y’(t) + 13y(t) = 40 sin(3t).
Chomcierislic QCL“ ml‘flm + )3 =<m ‘lelq =0
30 BAﬂC'l'IOH lg \ﬁef 0162 C03 +961ng C’MQ331 ‘ Lj’f Asm at “t B COSZ‘t no overqu.
3?‘ = 73A cosy: %B 3m3+
3P“ =‘q/5rsm3l: cc] Ecog3t'
Substiluie
(31A H15 + BA) 3m 3t +( 4115 41/4 + 13133 COS gt =40sm’5t
(4A +ms )smgt + (—m +453 Cos at = 40sm3'l; 2. [5 points] Find the general solution to the inhomogeneous second order constant
coefficient ordinary differential equation y”(t) + y’(t) — 2y(t) = 4e‘ + 5e2t Chomdeﬁs’li c cg" ml+m —2 : (mi1) (rm—D =0
~ ‘ v ~21: +t
So comylcmm‘l'cinj limchon IS ch=C‘ e, ‘t (116 , Guess (A?) :AeJC . Overlays 3o molhplg
3F, 2 Atct no ovaqu.
9P: 2 Ate:t +Ae’C
«a? = Atet +2Aejc' Subsl‘l'iuie .
‘3)“: el’ =— Lick =§ A=+/3 Quqsﬁom "U cjrok 4A +11 B =40 Equah‘ng wemcienh
'vIm + 4r rs "=0 lmmgdwojrelj we 522 i‘ho’r B=$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 nI 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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This note was uploaded on 04/30/2009 for the course M 58055 taught by Professor Windsor during the Spring '07 term at University of Texas at Austin.
 Spring '07
 Windsor
 Calculus

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