; TeX output 1996.03.12:1143덍䍍W"V cmbx1018.01GPR9OBLEMSET4 ;HandTintoRoQom2-108b9y1:00pmF riday,TMarch22.!9I9Zffƶ ffD Ǎ!p0J cmsl10The3duedateandtimeareabsolute.i`Latehomeworksareaccepted untilUUthesolutionsarepGosted,butthescorewillbemultipliedbyK`y cmr101 b> cmmi10=2.EverythingMincludedheremaybGeusedlater.Y*oudonotneedtomemorizeUUtheproGofs,butyouneedtoknowthefacts.ffDD9ZffM􍍑 11.9MVT.1/3,UU5.q135/10,18,19.141/21,24.144/6,11,19. Fd(10UUpGoints) 12.9Inothisexercisewesolvethedi erentialequationf^O!cmsy70aȟ^00(x)C=f(x).Inootherwords9weQareloGokingforfunctionswhicharetheirownsecondderivqatives.pAsawarm9up,UU rst ndallofthoseg.functionswhichsatisfyg[ٟ^0*(x)=g[٫(x).r(a)9SuppGoseUUg[ٟ^0*(x)=g[٫(x).qProvethatthereexistsacٓRcmr72ȫconstantsuchthat'g[٫(x)=c1S !", cmsy108e 0ercmmi7x:䛫(b)9SuppGoseUUh^09(x)=h(x).qProvethatthereexistsac2ȫconstantsuchthath(x)=c2S8ex H: (c)9SuppGoseUUf^0aȟ^00(x)=f(x).qProvethatthereexistc1|s;c2ȫconstantssuchthatRɱf(x)8+f0ȫ(x)=c1S8exand,zf(x)f0ȫ(x)=c2S8ex H:䛫(d)9ConcludeUUthatthereexista;bconstantssuchthatf(x)=a8exAī+bex H:9$': cmti10R}'emark.pNotethatsinh'xandcoshD2xaretheirownsecondderivqativesandindeed 9theyUUareamongthesolutions,toGo.Fd(10UUpGoints)#N|{Ycmr8TÎyp cmmi10ٓRcmr7|{Ycmr8K`y cmr10