WPCOE ]WZG[MU3 = bv?U . f`N n5u xz3E^VV"~]D0WMɅwA}N`&Z1Mij 񥶟n7Nc ׳ ^|ڒߑ.1ɊI831`>9{:,4kqwAeD.^waQ9ȅ 8tR.quGO р[i]dwaʲʶf #[זEMRglSyLSG*2L-L-}i;]VfS׿bMʤ@4 Ik\ ӫqSA?iЮ(~yR'.;#!:UN[ % 0(U:c ) @ oT  pb'Yw@;4;<< m <UN7<]<P<D2>Lv>I?L @SWAPAAAAAAAAAAAAA 0DBB AS>C>C>C AC D+:D 0KeDeDeDeDeDeDeDeDeD AYDD 0F EHP DeskJet 500,X,,,,,X0x9 Z6Times New Roman RegularX($<6X9`("Courier 10cpiLevel 1Level 2Level 3Level 4Level 5Level 1Level 2Level 3Level 4Level 5Level 1Level 2Level 3Level 4Level 5Level 1Level 2Level 3Level 4Level 5Level 1Level 2Level 3Level 4Level 5 *OLE*WPCS hg 03'XXXX,,,,51391 ! g  "XX #Xr. $e. ,'1 &cqd9 Z6Times New Roman RegularcXXX        @  2"'cqd9 Z6Times New Roman RegularcXXX        @  1"(@cqd9 Z6Times New Roman RegularcXXX        @  3") 9cqd9 Z6Times New Roman RegularcXXX        @  4"* 9cqd9 Z6Times New Roman RegularcXXX        @  4"+?" cqd9 Z6Times New Roman RegularcXXX        @  5",}acqd9 Z6Times New Roman RegularcXXX        @  6'1 .nXX / ,6XWPWin 6.0/OLE 1.0 Prefix Information MarkerWPDraw30.Drawingࡱ>   Root Entry`.@p@cnOle 7WPG20T CompObj'n  !"#$%&(+,-./0123456789:;<=>?@ABCDEFGHIJKLM'F! WP 7 [0]WPCS hg 03'XXXX,,,,51391 ! g  "XX #Xr. $e. ,'1 &cqd9 Z6Times New Roman RegularcXXX        @  2"'cqd9 Z6Times New Roman RegularcXXX        @  1"(@cqd9 Z6Times New Roman RegularcXXX        @  3") 9cqd9 Z6Times New Roman RegularcXXX        @  4"* 9cqd9 Z6Times New Roman RegularcXXX        @  4"+?" cqd9 Z6Times New Roman RegularcXXX        @  5",}acqd9 Z6Times New Roman RegularcXXX        @  6'1 .nXX / ,6X `.@pPresentations 7 Drawing Embed SourceWPDraw30.Drawing9qPP  " f  " """DDDUUUwww@3f3333f333ff3fffff3f3f̙3f3333f3333333333f333333BOlePart)OlePres000*3f3f33ff3f3f3f3333f3333333f3̙333333f333ff3ffffff3f33f3ff3f3f3ffff3fffffffffff3fffffff3fff̙ffff3fffff3f3333f333ff3fffff3f3f̙3f3f3333f333ff3fffff̙̙3̙f̙̙̙3f̙3f3f3333f333ff3fffff3f3f̙3f45  !Q "- >>-- *--^x--x-Times New RomanL-- .  "Systemn-- 2 |32|;-- .  -- 2 Z1b-- .  -- 2 3-- .  -- 2 4-- .  -- 2 4-- .  -- 2 5-- .  -- 2 26:--(H--8- METAFILEPICT"<" f  " """DDDUUUwww@3f3333f333ff3fffff3f3f̙3f3333f3333333333f3333333f3f33ff3f3f3f3333f3333333f3̙333333f333ff3ffffff3f33f3ff3f3f3ffff3fffffffffff3fffffff3fff̙ffff3fffff3f3333f333ff3fffff3f3f̙3f3f3333f333ff3fffff̙̙3̙f̙̙̙3f̙3f3f3333f333ff3fffff3f3f̙3f45  !Q "- >>-- *--^x--x-Times New RomanL-- .  "Systemn-- 2 |32|;-- .  -- 2 Z1b-- .  -- 2 3-- .  -- 2 4-- .  -- 2 4-- .  -- 2 5-- .  -- 2 26:--(H--8- ,- .345E6789-i$:i+003|x9U(9 Z 6Times New Roman Regular Sy`(t)``=``alpha``cos``(omega``t)```+```{beta``sin``(omega``t)}OVERomegaXXyXX(XXtXX)XXwXXXXDcosXX|(XX3XXxtXX)XX6{XXXXXX8XsinXX@X(XXX3XX<XtXXtX)XXG3 :INT_{INF}^INF```delta`(`t``B)``f`(t)```dt``=``f`(B) RLn>>w  (qt?B)Jf(t/)dt=f(Bn) ?INT_{INF}^INF```delta`(`t``B)``f`(t)```dt``=``f`(B) RLn>>w  (qtB )f(/to)dt}f("B) Iy`(t)``=``alpha``cos``(omega``t)```+```{beta``sin``(omega``t)}OVERomegaXXyXXt(XXtXX)XX`XXXX-cosXXe(XX3XXatXX)XX{XoXXXXX!XsinXX)X(XXmX3XX%XtXX]X)XXG3(3$ !  (M -, dc(/.&OLE 2.0 Box <=8C HKKKK'dxd( $ Figure  1   (:.M << deUU(n$ (  1  ) !  ݛXXXXTakeHomePowerSeriesSolutionofOrdinaryDifferential  EquationsFindageneralsolutionto@"2y"+xy'+y=xintheformofapowerseriesabouttheordinarypointx=0.Due:TuesdayMay3,1994#XaXXX#     InclassClosedbookportion.0  0h(#(#0h(#h(#0(#(#0p(#(#Name:_________________p(#p(# 1)Findthepowerseriessolutionabout66x=077toy'+2xy=0.2)UseLaplaceTransformstosolve:@y"y=g(t),with88y(0)=1,y'(0)=2where   `    1fort<3  8     `  g(t)=  $     `    tfort>3   3.SolvetheEulerequation2x2y"+5xy'+y=0   4.Writethefunctiongiveninthegraphusingtheunitstepfunctionm- 9)%`|0.o `0*xEF  6m (#(#(#(#Allofthepiecesarestraightlines. $X"(#(# -XaXXXa -3OpenBook/Notes/Computer34Test43  3-Background:   AspringwithmassmandspringconstantksatisfiesHooke'sLaw.Ifthevariableymeasuresthedisplacementfromtheequilibrium,andtheinitialpositionandvelocityofthemassaregivenbyy(0)=withy'(0)=,thenysatisfiesthedifferentialequation 8     `     h   my"+ky=0withinitialconditionsy(0)=withy'(0)=.  ` If32=k/m,thenthedifferentialequationcanberewritteny"+32y=0andinitialconditions  8  ̀y(0)=withy'(0)=.(ThesolutiononMapleis-9-W*! `EXvW-99W*! `ESSW9-.)   Ifaforcingfunctionisintroduced,thenthedifferentialequationbecomesy"+32y=f(t).    Problems:  p  1. Solvey"+32y=0withI.C.y(0)=withy'(0)=𣀀onMaple. H  2. 0  a) Solvey"+32y=A (tB)withI.C.y(0)=withy'(0)=onMaple.Here isthe ` ТDiracdeltafunction. (#(# 0  b) RewriteyoursolutionintermsofsinesandcosineswhichinvolveonlytheconstantsA, ( B,,,and3.Hint:Youmaywanttousethepropertyof that(#(# -.-e8/` 0  `Ex< }e-..e8/` 0  `Ex d  e (#(#    (#(#.- 3. Inproblem#2,thespringmasssystemisstartedattimet=0withinitialpositiony(0)=and P! initialspeedy'(0)=.Attimet=BitisstruckbyablowofmagnitudemA. @"    a) FindthesolutiontotheIVPy"+y=3 (t!)withI.C.y(0)=1withy'(0)=1. $h"   b) Graphthesolutiony(t)anditsderivativey'(t)onthesameaxes. %D!$  4. Thespringmasssystemcanactuallybestoppedbyasingleblowaswell.FindthevalueforA ' #& sothatthespringisstoppedafteratimet=B(i.e.y(t)=0fort>B).Inotherwords,findAandBsothaty(t)=0fort>Bisasolutionto  y"+y=A (tB)withI.C.y(0)=1,y'(0)=1. +&* -5--- \-(, -5-#XaXXXap#-,Computer+:+,5++5