loupe (anl2275) – H.W. 9.1 – wolesensky – (53050)1Thisprint-outshouldhave7questions.Multiple-choice questions may continue onthe next column or page – find all choicesbefore answering.00110.0pointsFind all nonzero values ofkfor which thefunctiony= sinktsatisfies the differentialequationy′′+ 9y= 0.1.k= 3,-3correct2.k= 9,-93.k=-34.k= 95.k= 36.k=-9Explanation:We will begin by solving fory′′.y= sinkty′=kcoskty′′=-k2sinkt.We computey′′+ 9y=-k2sinkt+ 9 sinkt= (9-k2) sinkt.Hencey′′+ 9y= 0 if and only ifk2= 9, i.e., ifand only ifk=±3. Hence,k= 3,-3.00210.0pointsFind all values ofrfor which the functiony=ertsatisfies the differential equationy′′-2y′-48y= 0.1.r=-62.r= 483.r=-48,-24.r=-8,65.r=-6,8correct6.r= 2,48Explanation:We will begin by solving fory′andy′′.y=erty′=rerty′′=r2ert.We computey′′-2y′-48y=r2ert-2rert-48ert= (r2-2r-48)ert= (r+ 6)(r-8)ert.Hencey′′-2y′-48y= 0 if and only ifk=-6,8. Thus,k=-6,8.00310.0pointsWhich of the following functions satisfy thedifferential equationy′′+ 4y′+ 4y= 0 ?1.y=e2t, te2t2.y=e−2t, te−2tcorrect3.y=e2t, te4t4.y=e−2t, te2t

loupe (anl2275) – H.W. 9.1 – wolesensky – (53050)25.y=e−4t, te−2t6.y=e−2t, te−4t