Solutions Manual for Advanced Engineering Mathematics with MATLAB 5th Edition - Duffy (Complete Solutions)

All Chapters Covered
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SOLUTIONS

,TableofContents j j




Chapter 1: First- j j




Order Ordinary Differential Equations 1 Chapter2:Higher-
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OrderOrdinaryDifferentialEquations Chapter 3: Linear Al
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gebra
Chapter 4: Vector Calculus Chapte
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r 5: Fourier Series Chapter 6: The F
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ourier Transform j




Chapter7:TheLaplace Transform Cha
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pter 8: The Wave Equation Chapter 9:
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The Heat Equation Chapter 10: Laplac
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e’s Equation
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Chapter 11: The Sturm- j j j




Liouville Problem Chapter 12: Special Functio
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ns
Appendix A: Derivation of the Laplacian in Polar Coordinates Appendix B:
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Derivation of the Laplacian in Spherical Polar Coordinates
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, Solution Manual j




Sectionj 1.1

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

Sectionj 1.2

1. Becausejthejdifferentialjequationjcanjbejrewrittenje−yjdyj=jxdx,jintegra-
jtionjimmediately jgives j—e− j=j 2jx j—jC,jorjyj=j—jln(Cj—jx /2).
y 1 2 2



2. Separatingjvariables,jwejhavejthatjdx/(1j+jx2)j=jdy/(1j+jy2).jIntegratingjthi
— − (y)j=jtan(C),jorj(xj y)/(1+xy)
sjequation,jwejfindjthatjtan−1(x)j tan — j=jC.
1



3. Becausejthejdifferentialjequationjcanjbejrewrittenjln(x)dx/xj=jyjdy,jinte-
2j jln (x)j+jCj=j jy2 ,jorjy (x)j—jln (x)j=j2C.
1 2 1 2 2 2
jgrationjimmediatelyjgives



4. Becausej thej differentialj equationj canj bej rewrittenj y2jdyj =j (xj+jx3)jdx,jint
egrationjimmediatelyjgivesjy3(x)/3j=jx2/2j+jx4/4j+jC.

5. Becausej thej differentialj equationj canj bej rewrittenj yj dy/(2+y2)j =j xdx/(1+
x2),jintegrationj immediatelyj givesj 1jln(2j+jy2)j=j 1jln(1j+jx2)j+j1jln(C),j or
2 2 2
2j+jy2(x)j=jC(1j+jx2).

6. Becausej thej differentialj equationj canj bej rewrittenj dy/y1/3j =j x1/3jdx,jint
j j 3/2
1j 4/3
egrationjimmediatelyjgivesj 3jy2 2/3j =j 3jx4/3 3
4 j+j jC,jor
2 jy(x)j= x j+jC2 .

1

, 2 Advancedj Engineeringj Mathematicsj withj MATLAB

7. Becausejthejdifferentialjequationjcanjbejrewrittenje−yjdyj=jexjdx,jintegra-
jtionjimmediatelyjgivesj—e− j=je j—jC,jorjy(x)j=j—jln(Cj—je ).
y x x


8. Becausej thej differentialj equationj canj bej rewrittenj dy/(y2j+j1)j =j (x3j+j5)jd
x,j integrationj immediatelyj givesj tan−1(y)j =j 1jx4j+j5xj+jC,j orj y(x)j =
j j 4
tan 14jx4j+j5xj+jC .

9. Becausej thej differentialj equationj canj bej rewrittenj y2jdy/(bj—jay3)j =j dt,
y
integrationj immediatelyj givesj ln[bj—jayj ]j3y0jj =j —3at,j orj (ay 3 3
—jb)/(ay0j—jb)j=
e −3at. j



10. Becausejthejdifferentialjequationjcanjbejwrittenjdu/uj=jdx/x2,jintegra-
jtionjimmediatelyj givesjuj=jCe
−1/xj orjy(x)j=jxj+jCe−1/x.



11. Fromj thej hydrostaticj equationj andj idealj gasj law,j dp/pj= —
gjdz/(RTj).jSubstitutingjforjTj(z),
dp g
=j—j dz.
pj R(T 0 —jΓz)
Integratingjfromj0jtojz,
j
T0j—jΓzj g/(RΓ)
j j
p(z) j gj j j
p(z)
ln T0j—
j
ln = , or = .
p0 RΓ jΓzjT0 p0 T0


12. Forj 0j <j zj <j H,j wej simplyj usej thej previousj problem.j Atj zj =j H,j thejpres
surejis
j
T0j—jΓHj jg/(RΓ)
p(H)j =j p0 .
T0
Thenj wej followj thej examplej inj thej textj forj anj isothermalj atmospherej for
zj≥jH.

13. Separatingj variables,j wej findj that
dV dV RjdV dt
=j —j =j—j .
Vj +jRVj2/Sj Vj j S(1j+jRV/S)j RCj

Integrationj yields
j j j
V j tj j
ln 1j+jRV/S =j—jRC +jln(C).

Uponj applyingj thej initialj conditions,

V0 j −t/(RC)j j j RV0/Sj j
Vj(t)j=j e +j e−t/(RC)Vj(t).
1j+jRV0/S 1j+jRV0/S