solutions manual to Advanced Engineering Mathematics with MATLAB, 5th Edition. Duffy

All Chapters Covered
m m




SOLUTIONS

,Table of Contents
m m




Chapterm1:mFirst-
OrdermOrdinarymDifferentialmEquationsm1mChapterm2:mHi
gher-
OrdermOrdinarymDifferentialmEquationsmChapterm3:mLine
armAlgebra
Chapterm4:mVectormCalculusmCha
pterm5:mFouriermSeriesm Chapterm6
:mThemFouriermTransform
Chapterm7:mThemLaplacemTransformm
Chapterm8:mThemWavemEquationmCh
apterm9:mThemHeatmEquationmChapt
erm10:mLaplace’smEquation
Chapterm11:mThemSturm-
LiouvillemProblemmChapterm12:mSpecialmFun
ctions
AppendixmA:mDerivationmofmthemLaplacianminmPolarmCoordinatesmAppe
ndixmB:mDerivationmofmthemLaplacianminmSphericalmPolarmCoordinates

, Solution Manual m




Sectionm 1.1

1.m first-order,m linear 2.m first-order,m nonlinear
3.m first-order,m nonlinear 4.m third-order,m linear
5.m second-order,m linear 6.m first-order,m nonlinear
7.m third-order,m nonlinear 8.m second-order,m linear
9.m second-order,m nonlinear 10.m first-order,m nonlinear
11.m first-order,m nonlinear 12.m second-order,m nonlinear
13.m first-order,m nonlinear 14.m third-order,m linear
15.m second-order,m nonlinear 16.m third-order,m nonlinear

Sectionm 1.2

1. Becausemthemdifferentialmequationmcanmbemrewrittenme−ymdym=mxdx,min
tegra-mtionmimmediatelymgives2m—e−ym=m 1mx2m—mC,mormym=m—mln(Cm—
2
mx /2).


2. Separatingmvariables,mwemhavemthatmdx/(1m+mx2)m=mdy/(1m+my2).mInt
egratingmthismequation,mwemfind—
mthat mtan− (x)m tan− (y)m—
1 1
=mtan(C),mor
m(xm y)/(1+xy)m=mC.



3. Becausemthemdifferentialmequationmcanmbemrewrittenmln(x)dx/xm=mymdy
mgivesm mln2 (x)m+mCm= m my ,mormy (x)m—
1 2 1 2 2
,minte-mgrationmimmediately
2
2
mln (x)m=m2C.


4. Becausem them differentialm equationm canm bem rewrittenm y2mdym =m (xm+
3 3 2 4
mx )mdx,mintegrationmimmediately mgivesmy (x)/3m=m x /2m+mx /4m+mC.



5. Becausem them differentialm equationm canm bem rewrittenm ym dy/(2+y2)m =m xdx/(1+
x2),mintegrationm immediatelym givesm 1mln(2m+my2)m=m 1mln(1m+mx2)m+m1mln(C),
m or
2 2 2
2m+my2(x)m=mC(1m+mx2).

6. Becausem them differentialm equationm canm bem rewrittenm dy/y1/3m =m x
1/3 3 2/3 3 4/3 3
mdx,mintegrationmimmediately mgivesm my m =m mx m+m mC,mormy(x)m
m m 3/2 2 4 2 2
1m 4/3
= x m+mC .

1

, 2 Advancedm Engineeringm Mathematicsm withm MATLA
B
7. Becausemthemdifferentialmequationmcanmbemrewrittenme−ymdym=mexmdx,mi
ntegra-mtionmimmediatelymgivesm—e−ym=mexm—mC,mormy(x)m=m—
mln(Cm— me ).
x


8. Becausem them differentialm equationm canm bem rewrittenm dy/(y2m+m1)m
=m (x3m+m5)mdx,m integrationm immediatelym givesm tan−1(y)m =m 1mx4m+m5
xm+mC,m orm y(x)m =
m m 4
tan 4 1mx4m+m5xm+mC .

9. Becausem them differentialm equationm canm bem rewrittenm y2mdy/(bm—may3)m =m dt,
3 y
integrationm immediatelym givesm ln[bm—maymm ]m y0m =m —
3 3
—mb)/(ay0m—
3at,m orm (ay mb)m=
e−3at. m


10. Becausemthemdifferentialmequationmcanmbemwrittenmdu/um=mdx/x2,min
tegra-mtionmimmediately m givesmum=mCe−1/xm ormy(x)m=mxm+mCe−1/x.

— m=
11. Fromm them hydrostaticm equationm andm idealm gasm law,m dp/p
gmdz/(RTm).mSubstitutingmformTm(z),
dp g
=m— dz.
pm 0 —mΓz)
m
Integratingmfromm0mto R(T
mz,
m

m m
p(z) gm p(z) T0m—mΓzm g/(RΓ)
lnmT0m—
m
ln = , or = .
p0 RΓ mΓzm p0 m
T0
m
T0


12. Form 0m <m zm <m H,m wem simplym usem them previousm problem.m Atm zm
=m H,m thempressuremis
m
T0m—mΓHm g/(RΓ)
p(H)m =m p0 .
T0
m
Thenm wem followm them examplem inm them textm form anm isothermalm atmospherem for
zm≥mH.

13. Separatingm variables,m wem findm that
dV dV RmdV dt
=m —m =m—m .
Vm +mRVm2/Sm Vm m S(1m+mRV/S)m RCm

Integrationm yields

m mm
V m tm m
ln 1m+mRV/ =m—mRC +mln(C).
S
Uponm applyingm them initialm conditions,

V0 m −t/(RC)m m m RV0/Sm m −t/(RC)
Vm(t)m=m e +m e Vm(t).
1m+mRV0/S 1m+mRV0/S