PAPER2:纸2

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MATHEMATICALTRIPOSPartIA

Friday1June20011.30to4.30

PAPER2

Beforeyoubeginreadtheseinstructionscarefully.

EachquestioninSectionIIcarriestwicethecreditofeachquestioninSectionI.You

mayattemptallfourquestionsinSectionIandatmostfivequestionsfromSection

II.InSectionIInomorethanthreequestionsoneachcoursemaybeattempted.

Completeanswersarepreferredtofragments.

Writeononesideofthepaperonlyandbegineachansweronaseparatesheet.

Writelegibly;otherwiseyouplaceyourselfatagravedisadvantage.

Attheendoftheexamination:

Tieupyouranswersintwobundles,markedBandFaccordingtothecodeletter

affixedtoeachquestion.Attachabluecoversheettoeachbundle;writethecodein

theboxmarked‘SECTION’onthecoversheet.DonottieupquestionsfromSection

IandSectionIIinseparatebundles.

Youmustalsocompleteagreenmastercoversheetlistingallthequestionsattempted

byyou.

Everycoversheetmustbearyourexaminationnumberanddesknumber.2

SECTIONI

1BDifferentialEquations

Findthesolutionto

dy(x)

dx+tanh(x)y(x)=H(x),

intherange−∞

definedby

H(x)=󰀇0x<0

1x>0.

Sketchthesolution.

2BDifferentialEquations

Thefunctiony(x)satisfiestheinhomogeneoussecond-orderlineardifferential

equation

y󰀂󰀂−y󰀂−2y=18xe−x.

Findthesolutionthatsatisfiestheconditionsthaty(0)=1andy(x)isboundedasx→∞.

3FProbability

ThefollowingproblemisknownasBertrand’sparadox.Achordhasbeenchosen

atrandominacircleofradiusr.Findtheprobabilitythatitislongerthanthesideof

theequilateraltriangleinscribedinthecircle.Considerthreedifferentcases:

a)themiddlepointofthechordisdistributeduniformlyinsidethecircle,

b)thetwoendpointsofthechordareindependentanduniformlydistributedover

thecircumference,

c)thedistancebetweenthemiddlepointofthechordandthecentreofthecircle

isuniformlydistributedovertheinterval[0,r].

[Hint:drawingdiagramsmayhelpconsiderably.]

Paper23

4FProbability

TheRuritanianauthoritiesdecidedtopardonandreleaseoneoutofthreeremaining

inmates,A,BandC,keptinstrictisolationinthenotoriousAlkazafprison.Theinmates

knowthis,butcan’tguesswhoamongthemistheluckyone;thewaitingisagonising.A

sympathetic,butcorrupted,prisonguardapproachesAandofferstoname,inexchange

forafee,anotherinmate(notA)whoisdoomedtostay.Hesays:“Thisreducesyour

chancestoremainherefrom2/3to1/2:willitmakeyoufeelbetter?”Ahesitatesbut

thenacceptstheoffer;theguardnamesB.

AssumethatindeedBwillnotbereleased.Determinetheconditionalprobability

P󰀋Aremains󰀂󰀂Bnamed󰀁=P(A&Bremain)P(Bnamed)

andthuschecktheguard’sclaim,inthreecases:

a)whentheguardiscompletelyunbiased(i.e.,namesanyofBandCwith

probability1/2ifthepairB,Cistoremainjailed),

b)ifhehatesBandwouldcertainlynamehimifBistoremainjailed,

c)ifhehatesCandwouldcertainlynamehimifCistoremainjailed.

Paper2[TURNOVER4

SECTIONII

5BDifferentialEquations

Therealsequenceyk,k=1,2,...satisfiesthedifferenceequation

yk+2−yk+1+yk=0.

Showthatthegeneralsolutioncanbewritten

yk=acosπk

3+bsinπk

3,

whereaandbarearbitraryrealconstants.

Nowletyksatisfy

yk+2−yk+1+yk=1

k+2.(∗)

Showthataparticularsolutionof(∗)canbewrittenintheform

yk=k󰀉

n=1ank−n+1,

where

an+2−an+1+an=0,n≥1,

anda1=1,a2=1.

Hence,findthegeneralsolutionto(∗).

Paper25

6BDifferentialEquations

Thefunctiony(x)satisfiesthelinearequation

y󰀂󰀂(x)+p(x)y󰀂(x)+q(x)y(x)=0.

TheWronskian,W(x),oftwoindependentsolutionsdenotedy1(x)andy2(x)isdefined

tobe

W(x)=󰀂󰀂󰀂󰀂y1y2y1󰀂y2󰀂󰀂󰀂󰀂󰀂.

Lety1(x)begiven.Inthiscase,showthattheexpressionforW(x)canbe

interpretedasafirst-orderinhomogeneousdifferentialequationfory2(x).Hence,by

explicitderivation,showthaty2(x)maybeexpressedas

y2(x)=y1(x)󰀊x

x0W(t)

y1(t)2dt,(∗)

wheretherˆoleofx0shouldbebrieflyelucidated.

ShowthatW(x)satisfies

dW(x)

dx+p(x)W(x)=0.

Verifythaty1(x)=1−xisasolutionof

xy󰀂󰀂(x)−(1−x2)y󰀂(x)−(1+x)y(x)=0.(†)

Hence,using(∗)withx0=0andexpandingtheintegrandinpowersofttoordert3,find

thefirstthreenon-zerotermsinthepowerseriesexpansionforasolution,y2(x),of(†)

thatisindependentofy1(x)andsatisfiesy2(0)=0,y2󰀂󰀂(0)=1.

Paper2[TURNOVER6

7BDifferentialEquations

Considerthelinearsystem

˙z+Az=h,(∗)

where

z(t)=󰀅x(t)

y(t)󰀆

,A=󰀅1+a−2

1−1+a󰀆

,h(t)=󰀅2cost

cost−sint󰀆

,

wherez(t)isrealandaisarealconstant,a≥0.

Finda(complex)eigenvector,e,ofAanditscorresponding(complex)eigenvalue,

l.Showthatthesecondeigenvectorandcorrespondingeigenvaluearerespectively¯eand¯l,wherethebaroverthesymbolssignifiescomplexconjugation.Henceexplainhowthe

generalsolutionto(∗)canbewrittenas

z(t)=α(t)e+¯α(t)¯e,

whereα(t)iscomplex.

Writedownadifferentialequationforα(t)andhence,fora>0,deducethesolution

to(∗)whichsatisfiestheinitialconditionz(0)=0¯.

Isthelinearsystemresonant?

Bytakingthelimita→0ofthesolutionalreadyfounddeducethesolution

satisfyingz(0)=0¯whena=0.

8BDifferentialEquations

Carnivoroushuntersofpopulationhpreyonvegetariansofpopulationp.Inthe

absenceofhuntersthepreywillincreaseinnumberuntiltheirpopulationislimitedby

theavailabilityoffood.Intheabsenceofpreythehunterswilleventuallydieout.The