Homework 05-solutions.pdf - ugurel(eu2243 Homework 05...

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ugurel (eu2243) – Homework 05 – staron – (53550)1Thisprint-outshouldhave18questions.Multiple-choice questions may continue onthe next column or page – find all choicesbefore answering.00110.0pointsFind all values ofkthat don’t result in a zerofunction for which the functiony= sinktsatisfies the differential equationy′′+ 9y= 01.k= 9,92.k=33.k= 3,3correct4.k=95.k= 96.k= 3Explanation:We will begin by solving fory′′.y= sinkty=kcoskty′′=k2sinkt.We computey′′+ 9y=k2sinkt+ 9 sinkt= (9k2) sinkt.Hencey′′+ 9y= 0 if and only ifk2= 9, i.e., ifand only ifk=±3. Thus,k= 3,3.00210.0pointsFind all nonzero values ofkfor which thefunctiony=Asinkt+Bcosktsatisfies thedifferential equationy′′+ 36y= 01.k= 362.k= 36,363.k= 6,6correct4.k=65.k=366.k= 6Explanation:We will begin by solving fory′′.y=Asinkt+Bcoskty=AkcosktBksinkty′′=Ak2sinktBk2coskt=k2(Asinkt+Bcoskt)=k2y.We computey′′+ 36y=k2y+ 36y= (36k2)y.Hencey′′+36y= 0 if and only ifk2= 36, i.e.,if and only ifk=±6. Hence,k= 6,6.00310.0pointsFind all values ofrfor which the functiony=ertsatisfies the differential equationy′′4y12y= 0.1.r= 122.r=6,2
for all values ofAandB.
ugurel (eu2243) – Homework 05 – staron – (53550)23.r=12,44.r= 4,125.r=26.r=2,6correctExplanation:We will begin by solving foryandy′′.y=erty=rerty′′=r2ert.We computey′′4y12y=r2ert4rert12ert= (r24r12)ert= (r+ 2)(r6)ert.Hencey′′4y12y= 0 if and only ifk=2,6. Thus,k=2,6.00410.0pointsWhich of the following functions satisfy thedifferential equationy′′+ 10y+ 25y= 0 ?1.y=e5t, te5t2.y=e5t, te5t3.y=e5t, te5tcorrect4.y=e5t, te10t5.y=e5t, te10t6.y=e10t, te5tExplanation:All answer choices above are of the formertandtert, so let us solve for potential valuesofrin each of those cases. We will begin byassumingy=ertand findingyandy′′.y=erty=rerty′′=r2ert.We computey′′+ 10y+ 25y=r2ert+ 10rert+ 25ert= (r2+ 10r+ 25)ert= (r+ 5)2ertHencey′′+10y+25y= 0 if and only ifr=5.This meanse5twill satisfy this equation.Now we will assumey=tertand findyandy′′.y=terty=rtert+ert= (rt+ 1)erty′′=rert+r2tert+rert=r2tert+ 2rert= (r2t+ 2r)ert.Now we computey′′+ 10y+ 25y= (r2t+ 2r)ert+ 10(rt+ 1)ert+ 25tert= (r2t+ 2r+ 10rt+ 10 + 25t)ert=bracketleftbig(r2+ 10r+ 25)t+ 2r+ 10bracketrightbigert=bracketleftbig(r+ 5)2t+ 2(r+ 5)bracketrightbigert= (r+ 5) [(r+ 5)t+ 2]ert.Hencey′′+ 10y+ 25y= 0 if and only if(r+ 5) [(r+ 5)t+ 2] = 0 for all values oft, soonlyr=5 solves the differential equation.Thereforey=te5tis a solution.Since in both casesr=5 solves the differ-ential equation, valid results for this problemincludey=e5tandy=te5t.
ugurel (eu2243) – Homework 05 – staron – (53550)300510.0pointsWhich of the following families of functions isthe solution to the differential equationy= 8xy?1.y=Cx182.y=Ce4x3.y=Ce4x2correct4.y=x8+C5.y=Ce8x2Explanation:This is a separable differential equation,

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