Solution Manual for Advanced Engineering Mathematics SI Edition 8t
m m m m m m m m
h Edition ONeil 1337274526 9781337274524
m m m m
Fullmlinkmdownload:
SolutionmManual:
m
mm
Second-
Order Di erential Equations
m m m
2.1 ThemLinearmSecond-OrdermEquation
1. Itmism amroutinemexerciseminmdimerentiationmtomshowmthatm y1(x)mandm y2(x)mare
solutionsmofmthemhomogeneousmequation,mwhilemyp(x)mismamsolutionmof
mthemnonhomogeneous mequation.mThemWronskianmofmy1(x) mandmy2(x)
mis
W m(x)m= 6mcos(6x)m m 6msin(6m x)m=m6msinm 2(x)m 6msin2(x)m=m6;
sin(6x) cos(6x)
andmthismismnonzeromformallmx,msomthesem solutionsmaremlinearlymindepe
ndentmonmthemrealmline.mThemgeneralmsolutionmofmthemnonhomogeneou
smdimerentialmequationmis
1
ym=mc1msin(6x)m+mc2mcos(6x)m+m36m(xm
1):mFormtheminitialmvaluemproblem,mwemneed
1
y(0)m=mc 36m =m5
2
somc2m=m179=36.mA
nd
m 1
y0(0)m=m2m=m6c1m+36
m
somc1m=m71=216.mThemuniquemsolutionmofmtheminitialmvaluemproblemmis
71 m 179 1
y(x)m 216 sin(6x) 36mcos(6x)m+m36m m (xm1):
=
2. ThemWronskianmofme4xmandme4m xmis
W m(x)m= 4e4x
4e m 4 mx =8 m 6=0
e4x e4 xm
39
©m2018mCengagemLearning.mAllmRightsmreserved.mMaymnotmbemscanned,mcopiedmormduplicated,mormpostedmtomampubliclymaccessiblemwebsite,minm
wholemorminmpart.
, 40 CHAPTERm2.m SECOND-ORDERmDIFFERENTIALmEQUATIONS
som thesem solutionsm ofm them associatedm homogeneousm equationm are
indepen-
dent.mWithmthemparticularmsolutionmyp(x)mofmthemnonhomogeneousmequ
ation,mthismequationmhasmgeneralmsolution
1 2 1
y(x)m=mc1me4xm+mc2me4m x4
m m m m
32 x :
Frommtheminitialmconditionsmwemobtain
1
y(0)m=mc1m+mc2 32m m =m12
and
y0(0)m=m4c1 4c2m=m3:
Solvemthesemtomobtainmc1m=m409=64mandmc2m=m361=64mtomobtainmthemsolution
409m 4 361 4m x 1
m 1
2m m
y(x)m= x +m 64me 4m 32m:
64me x
3. Themassociatedmhomogeneousmequationmhasmsolutionsme2mxmand
xm
me .mTheirmWronskianmis
2xex
=me3xe
W(x)=2xx
2e e
andmthismismnonzeromformallmx.mThemgeneralmsolutionmofmth
emnonhomogeneousmdimerentialmequationmis
2 m xm xm
y(x)m=mc1e +mc2e +m m15m:
2
Formtheminitialmvaluemproblem,msolve
15
y(0)m=m3m= c1m+mc2m+m m 2
and
y0(0)m=m1m= 2cm 1 c2
tomgetmc1m=m23=2;mc2m=m22.mTheminitialmvaluemproblemmhasmsolution
23m2x x 15
y(x)m= 2me 22em + 2m:
4. Themassociatedmhomogeneousmequationmhasmsolutions
y1(x)m=me3xmcos(2x);my2(x)m=me3xmsin(2x):
ThemWronskianmofmthesemsolutionsmis
3xm
e cos(2x) e3xmsin(2x) =me6xm6=m0
W m(x)m=
3e3xmcos(2x)m 2e3xmsin(
2x)
©m2018mCengagemLearning.mAllmRightsmreserved.mMaymnotmbemscanned,mcopiedmormduplicated,mormpostedmtomampubliclymaccessiblemwebsite,minm
wholemorminmpart.
, 2.1. THEmLINEARmSECOND-ORDERmEQUATION 41
formallmx.mThemgeneralmsolutionmofmthemnonhomogeneousmequationmis
3x 3x 1mx
y(x)m=mc1e cos(2x)m+mc2e 8mem :
sin(2x)mTomsatisfymth
eminitialmconditions,mitmismrequiredmthat
1
y(0)m=m1m= c1
8
and
1
3c1m+m2c2 8m =m1:
Solvemthesemtomobtainmc1m=m7=8mandm c2m=m15=8.mThemsolutionmofmth
eminitialmvaluemproblemmis
y(x)m= 7me3xmcos(2x)m +m 15me3xm sin(2x) 1mex:
8 8 8
5. Themassociatedmhomogeneousmequationmhasmsolutions
y1(x)m=mexmcos(x);my2(x)m=mexmsin(x):
ThesemhavemWronskian
W m(x)m=m exmcos(x)m exmsin(x) ex sin(x)m+me cos(x)m=me2 6=m0
x x
exmcos(x) exmsin(x)
somthesemsolutionsmaremindependent.mThemgeneralmsolutionmof
mthemnonhomo-geneous mdimerentialmequationmis
5m 5
mm
y(x)m=mc1exmcos(x)m+mc2exmsin(x) 2 c2m 5x 2:
Wemnee
d 5
y(0)m=mc1 2m =m6
and
y0(0)m=m1m=mc1m+mc2 5:
Solvemthesemtomgetmc1m=m17=2mandmc2m=m5=2mtomgetmthemsolution
m17m xm m5 m m5m 2 m5m
y(x)m=m e cos(x) exmsin(x) x 5x :
2 2 2 2
6. Supposemy1mandmy2maremsolutionsmofmthemhomogeneousmequationm(2.2).mThen
y100m+mpy10m+mqy1m=m0
and
©m2018mCengagemLearning.mAllmRightsmreserved.mMaymnotmbemscanned,mcopiedmormduplicated,mormpostedmtomampubliclymaccessiblemwebsite,minm
wholemorminmpart.
, 0
00m
y2 +mpy2
©m2018mCengagemLearning.mAllmRightsmreserved.mMaymnotmbemscanned,mcopiedmormduplicated,mormpostedmtomampubliclymaccessiblemwebsite,minm
wholemorminmpart.
m m m m m m m m
h Edition ONeil 1337274526 9781337274524
m m m m
Fullmlinkmdownload:
SolutionmManual:
m
mm
Second-
Order Di erential Equations
m m m
2.1 ThemLinearmSecond-OrdermEquation
1. Itmism amroutinemexerciseminmdimerentiationmtomshowmthatm y1(x)mandm y2(x)mare
solutionsmofmthemhomogeneousmequation,mwhilemyp(x)mismamsolutionmof
mthemnonhomogeneous mequation.mThemWronskianmofmy1(x) mandmy2(x)
mis
W m(x)m= 6mcos(6x)m m 6msin(6m x)m=m6msinm 2(x)m 6msin2(x)m=m6;
sin(6x) cos(6x)
andmthismismnonzeromformallmx,msomthesem solutionsmaremlinearlymindepe
ndentmonmthemrealmline.mThemgeneralmsolutionmofmthemnonhomogeneou
smdimerentialmequationmis
1
ym=mc1msin(6x)m+mc2mcos(6x)m+m36m(xm
1):mFormtheminitialmvaluemproblem,mwemneed
1
y(0)m=mc 36m =m5
2
somc2m=m179=36.mA
nd
m 1
y0(0)m=m2m=m6c1m+36
m
somc1m=m71=216.mThemuniquemsolutionmofmtheminitialmvaluemproblemmis
71 m 179 1
y(x)m 216 sin(6x) 36mcos(6x)m+m36m m (xm1):
=
2. ThemWronskianmofme4xmandme4m xmis
W m(x)m= 4e4x
4e m 4 mx =8 m 6=0
e4x e4 xm
39
©m2018mCengagemLearning.mAllmRightsmreserved.mMaymnotmbemscanned,mcopiedmormduplicated,mormpostedmtomampubliclymaccessiblemwebsite,minm
wholemorminmpart.
, 40 CHAPTERm2.m SECOND-ORDERmDIFFERENTIALmEQUATIONS
som thesem solutionsm ofm them associatedm homogeneousm equationm are
indepen-
dent.mWithmthemparticularmsolutionmyp(x)mofmthemnonhomogeneousmequ
ation,mthismequationmhasmgeneralmsolution
1 2 1
y(x)m=mc1me4xm+mc2me4m x4
m m m m
32 x :
Frommtheminitialmconditionsmwemobtain
1
y(0)m=mc1m+mc2 32m m =m12
and
y0(0)m=m4c1 4c2m=m3:
Solvemthesemtomobtainmc1m=m409=64mandmc2m=m361=64mtomobtainmthemsolution
409m 4 361 4m x 1
m 1
2m m
y(x)m= x +m 64me 4m 32m:
64me x
3. Themassociatedmhomogeneousmequationmhasmsolutionsme2mxmand
xm
me .mTheirmWronskianmis
2xex
=me3xe
W(x)=2xx
2e e
andmthismismnonzeromformallmx.mThemgeneralmsolutionmofmth
emnonhomogeneousmdimerentialmequationmis
2 m xm xm
y(x)m=mc1e +mc2e +m m15m:
2
Formtheminitialmvaluemproblem,msolve
15
y(0)m=m3m= c1m+mc2m+m m 2
and
y0(0)m=m1m= 2cm 1 c2
tomgetmc1m=m23=2;mc2m=m22.mTheminitialmvaluemproblemmhasmsolution
23m2x x 15
y(x)m= 2me 22em + 2m:
4. Themassociatedmhomogeneousmequationmhasmsolutions
y1(x)m=me3xmcos(2x);my2(x)m=me3xmsin(2x):
ThemWronskianmofmthesemsolutionsmis
3xm
e cos(2x) e3xmsin(2x) =me6xm6=m0
W m(x)m=
3e3xmcos(2x)m 2e3xmsin(
2x)
©m2018mCengagemLearning.mAllmRightsmreserved.mMaymnotmbemscanned,mcopiedmormduplicated,mormpostedmtomampubliclymaccessiblemwebsite,minm
wholemorminmpart.
, 2.1. THEmLINEARmSECOND-ORDERmEQUATION 41
formallmx.mThemgeneralmsolutionmofmthemnonhomogeneousmequationmis
3x 3x 1mx
y(x)m=mc1e cos(2x)m+mc2e 8mem :
sin(2x)mTomsatisfymth
eminitialmconditions,mitmismrequiredmthat
1
y(0)m=m1m= c1
8
and
1
3c1m+m2c2 8m =m1:
Solvemthesemtomobtainmc1m=m7=8mandm c2m=m15=8.mThemsolutionmofmth
eminitialmvaluemproblemmis
y(x)m= 7me3xmcos(2x)m +m 15me3xm sin(2x) 1mex:
8 8 8
5. Themassociatedmhomogeneousmequationmhasmsolutions
y1(x)m=mexmcos(x);my2(x)m=mexmsin(x):
ThesemhavemWronskian
W m(x)m=m exmcos(x)m exmsin(x) ex sin(x)m+me cos(x)m=me2 6=m0
x x
exmcos(x) exmsin(x)
somthesemsolutionsmaremindependent.mThemgeneralmsolutionmof
mthemnonhomo-geneous mdimerentialmequationmis
5m 5
mm
y(x)m=mc1exmcos(x)m+mc2exmsin(x) 2 c2m 5x 2:
Wemnee
d 5
y(0)m=mc1 2m =m6
and
y0(0)m=m1m=mc1m+mc2 5:
Solvemthesemtomgetmc1m=m17=2mandmc2m=m5=2mtomgetmthemsolution
m17m xm m5 m m5m 2 m5m
y(x)m=m e cos(x) exmsin(x) x 5x :
2 2 2 2
6. Supposemy1mandmy2maremsolutionsmofmthemhomogeneousmequationm(2.2).mThen
y100m+mpy10m+mqy1m=m0
and
©m2018mCengagemLearning.mAllmRightsmreserved.mMaymnotmbemscanned,mcopiedmormduplicated,mormpostedmtomampubliclymaccessiblemwebsite,minm
wholemorminmpart.
, 0
00m
y2 +mpy2
©m2018mCengagemLearning.mAllmRightsmreserved.mMaymnotmbemscanned,mcopiedmormduplicated,mormpostedmtomampubliclymaccessiblemwebsite,minm
wholemorminmpart.
