Complete Solution Manual - Numerical Methods for Engineers, 7th Edition (Steven Chapra) – All Chapters 1-32 Included (Step-by-Step Verified A+ Guide)

1


CHAPTERV1
1.1 WeVwillVillustrateVtwoVdifferentVmethodsVforVsolvingVthisVproblem:V(1)VseparationVofVvariabl
es,VandV(2)VLaplaceVtransform.

dvV
=VgV−V
cV
vVdt m

SeparationVofVvariables:VSeparationVofVvariablesVgives


V
1
c

V
dvV=V VdtV
gV−V Vv
m

TheVintegralsVcanVbeVevaluatedVas
V cV 
lnV gV−V vV
 mV 

−
V
=VtV+
VCVcV/Vm

whereVCV=VaVconstantVofVintegration,VwhichVcanVbeVevaluatedVbyVapplyingVtheVinitialVconditionVtoVyield
V cV 
lnV gV−V v(0)V
 m 
CV = − V  
cV/Vm

whichVcanVbeVsubstitutedVbackVintoVtheVsolution
V cV  V cV 
lnV gV−V vV lnV gV−V v(0)V
 mV   m 
−  V =VtV−  
cV/Vm cV/Vm

ThisVresultVcanVbeVrearrangedValgebraicallyVtoVsolveVforVv,


vV=Vv(0)e−(c/Vm)tV +V
c
VmgV
(1−Ve −(cV/Vm)tV
)
whereVtheVfirstVpartVisVtheVgeneralVsolutionVandVtheVsecondVpartVisVtheVparticularVsolutionVforVth
eVconstantVforcingVfunctionVdueVtoVgravity.VForVtheVcaseVwhere,Vv(0)V=V0,VtheVsolutionVreducesV
toVEq.V(1.10)


vV = V
c
(
VmgV
1−Ve−(c/m)tV )
LaplaceVtransformVsolution:VAnValternativeVsolutionVisVprovidedVbyVapplyingVLaplaceVtransform
VtoVtheVdifferentialVequationVtoVgive

gV cV
sVV(s)V−Vv(0)V=V −V VV(s)
PROPRIETARYVMATERIAL.V ©VTheVMcGraw-
HillVCompanies,VInc.V AllVrightsVreserved.V NoVpartVofVthisVManualVmayVbeVdisplayed,VreproducedVorVdistri
butedVinVanyVformVorVbyVanyVmeans,VwithoutVtheVpriorVwrittenVpermissionVofVtheVpublisher,VorVusedVbey
ondVtheVlimitedVdistributionVtoVteachersVandVeducatorsVpermittedVbyVMcGraw-
HillVforVtheirVindividualVcourseVpreparation.V IfVyouVareVaVstudentVusingVthisVManual,VyouVareVusingVitVwi
thoutVpermission.

, 2

s m

SolveValgebraicallyVforVtheVtransformedVvelocity




PROPRIETARYVMATERIAL.V ©VTheVMcGraw-
HillVCompanies,VInc.V AllVrightsVreserved.V NoVpartVofVthisVManualVmayVbeVdisplayed,VreproducedVorVdistri
butedVinVanyVformVorVbyVanyVmeans,VwithoutVtheVpriorVwrittenVpermissionVofVtheVpublisher,VorVusedVbey
ondVtheVlimitedVdistributionVtoVteachersVandVeducatorsVpermittedVbyVMcGraw-
HillVforVtheirVindividualVcourseVpreparation.V IfVyouVareVaVstudentVusingVthisVManual,VyouVareVusingVitVwi
thoutVpermission.

, 3




VV(s)V = v(0) V
+
g (1)
sV+VcV/Vms(sV+VcV/Vm)

TheVsecondVtermVonVtheVrightVofVtheVequalVsignVcanVbeVexpandedVwithVpartialVfractions

g AV B
=V + (2)
A(sV+VcV/Vm)V+VBs
=V
s(sV+VcV/Vm) s sV+VcV/Vm s(sV+VcV/Vm)

ByVequatingVlikeVtermsVinVtheVnumerator,VtheVfollowingVmustVhold

g V = V AV 0V=V AsV+VBs
c
m

TheVfirstVequationVcanVbeVsolvedVforVAV=Vmg/c.VAccordingVtoVtheVsecondVequation,VBV=V–
A,VsoVBV=V–mg/c.VSubstitutingVtheseVbackVintoV(2)Vgives

g V=V
mgV/VcVV mgV/Vc
−V
s(sV+VcV/Vm) s sV+VcV/Vm

ThisVcanVbeVsubstitutedVintoVEq.V1VtoVgive

v(0) mgV/VcV mgV/Vc
VV(s)V = +V −V
sV+VcV/Vm s sV+VcV/Vm

TakingVinverseVLaplaceVtransformsVyields

mgV mgV −(cV/Vm)t
v(t)V =Vv(0)e−(c/Vm)tV +V +V e
c c

orVcollectingVterms

VmgV
v(t)V =Vv(0)e−(c/m)tV +V
c
(1−Ve −(c/Vm)tV
)
1.2 AtVtV=V8Vs,VtheVanalyticalVsolutionVisV41.137V(ExampleV1.1).VTheVrelativeVerrorVcanVbeVcalculatedVwith

analyticalV−Vnumerica
absoluteVrelativeVerr = 100%
l
or
analytical

TheVnumericalVresultsVare:

absolute
V Vstep
V V V Vv(8)
V V V V V V V V V r
elativeV error
2 44.8700 9.074%
1 42.8931 4.268%
0.5 41.9901 2.073%

PROPRIETARYVMATERIAL.V ©VTheVMcGraw-
HillVCompanies,VInc.V AllVrightsVreserved.V NoVpartVofVthisVManualVmayVbeVdisplayed,VreproducedVorVdistri
butedVinVanyVformVorVbyVanyVmeans,VwithoutVtheVpriorVwrittenVpermissionVofVtheVpublisher,VorVusedVbey
ondVtheVlimitedVdistributionVtoVteachersVandVeducatorsVpermittedVbyVMcGraw-
HillVforVtheirVindividualVcourseVpreparation.V IfVyouVareVaVstudentVusingVthisVManual,VyouVareVusingVitVwi
thoutVpermission.

, 4

TheVerrorVversusVstepVsizeVcanVthenVbeVplottedVas




PROPRIETARYVMATERIAL.V ©VTheVMcGraw-
HillVCompanies,VInc.V AllVrightsVreserved.V NoVpartVofVthisVManualVmayVbeVdisplayed,VreproducedVorVdistri
butedVinVanyVformVorVbyVanyVmeans,VwithoutVtheVpriorVwrittenVpermissionVofVtheVpublisher,VorVusedVbey
ondVtheVlimitedVdistributionVtoVteachersVandVeducatorsVpermittedVbyVMcGraw-
HillVforVtheirVindividualVcourseVpreparation.V IfVyouVareVaVstudentVusingVthisVManual,VyouVareVusingVitVwi
thoutVpermission.