Solution Manual of Differential Equations and

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1、1SolutionstoSection1.1True-FalseReview:1.FALSE.Aderivativemustinvolvesomederivativeofthefunctiony=f(x),notnecessarilythefirstderivative.2.TRUE.Theinitialconditionsaccompanyingadifferentialequationconsistofthevaluesofy,y0,...att=0.3.TRUE.Ifwedefinepositivevelocitytobeorienteddownward,thendv=g,dtwheregi

2、stheaccelerationduetogravity.4.TRUE.Wecanjustifythismathematicallybystartingfroma(t)=g,andintegratingtwicetoget1v(t)=gt+c,andthens(t)=gt2+ct+d,whichisaquadraticequation.25.FALSE.Therestoringforceisdirectedinthedirectionoppositetothedisplacementfromtheequilibriumposition.6.TRUE.AccordingtoNewton’sLa

3、wofCooling,therateofcoolingisproportionaltothedifferencebetweentheobject’stemperatureandthemedium’stemperature.Sincethatdifferenceisgreaterfortheobjectat100◦Fthantheobjectat90◦F,theobjectwhosetemperatureis100◦Fhasagreaterrateofcooling.7.FALSE.ThetemperatureoftheobjectisgivenbyT(t)=T+ce−kt,whereTisthe

4、temperaturemmofthemedium,andcandkareconstants.Sincee−kt6=0,weseethatT(t)6=Tforalltimest.Themtemperatureoftheobjectapproachesthetemperatureofthesurroundingmedium,butneverequalsit.8.TRUE.Sincethetemperatureofthecoffeeisfalling,thetemperaturedifferencebetweenthecoffeeandtheroomishigherinitially,duringthe

5、firsthour,thanitislater,whenthetemperatureofthecoffeehasalreadydecreased.9.FALSE.Theslopesofthetwocurvesarenegativereciprocalsofeachother.10.TRUE.Iftheoriginalfamilyofparallellineshaveslopeskfork6=0,thenthefamilyoforthogonaltra-jectoriesareparallellineswithslope−1.Iftheoriginalfamilyofparallellinesar

6、evertical(resp.horizontal),kthenthefamilyoforthogonaltrajectoriesarehorizontal(resp.vertical)parallellines.11.FALSE.Thefamilyoforthogonaltrajectoriesforafamilyofcirclescenteredattheoriginisthefamilyoflinespassingthroughtheorigin.Problems:22dydygt1.dt2=g=⇒dt=gt+c1=⇒y(t)=2+c1t+c2.Nowimposetheinitialc

7、onditions.y(0)=0=⇒2dygtc2=0.dt(0)=⇒c1=0.Hence,thesolutiontotheinitial-valueproblemis:qy(t)=2.Theobjecthitsthe2groundattime,t,wheny(t)=100.Hence100=gt0,sothatt=200≈4.52s,wherewehavetaken0020gg=9.8ms−2.d2y2.F