A First Course in Differential Equations 12th Edition – Zill – Solutions Manual

A First Course in Differential E
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quations with Modeling Appli
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cations, 12th Edition by Denni
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s G. Zill l l




Complete Chapter Solutions Manual are
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included (Ch 1 to 9) l l l l




** Immediate Download
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** Swift Response
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** All Chapters included
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,SolutionlandlAnswerlGuide:lZill,lDIFFERENTIALlEQUATIONSlWithlMODELINGlAPPLICATIONSl2024,l9780357760192;lChapterl#1:
IntroductionltolDifferentiallEquations




Solution and Answer Guide l l l


ZILL,lDIFFERENTIALlEQUATIONSlWITHlMODELINGlAPPLICATIONSl2024,l97
80357760192;lCHAPTERl#1:lINTRODUCTIONlTOlDIFFERENTIALlEQUATIONS


TABLE OF CONTENTSL L




Endl ofl Sectionl Solutions ........................................................................................................................................ 1
Exercisesl 1.1.......................................................................................................................................................... 1
Exercisesl 1.2........................................................................................................................................................14
Exercisesl 1.3........................................................................................................................................................22
Chapterl1linlReviewlSolutions ......................................................................................................................... 30




END OF SECTION SOLUTIONS
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EXERCISES 1.1 L



1. Secondl order;l linear
2. Thirdlorder;lnonlinearlbecauselofl(dy/dx)4
3. Fourthlorder;llinear
4. Secondlorder;lnonlinearlbecauseloflcos(rl+lu)
√
5. Secondlorder;lnonlinearlbecauselofl(dy/dx)2l or 1l+l(dy/dx)2
6. Secondlorder;lnonlinearlbecauseloflR2
7. Thirdlorder;llinear
8. Secondlorder;lnonlinearlbecauseloflẋ 2
9. Firstlorder;lnonlinearlbecauseloflsinl(dy/dx)
10. Firstlorder;llinear
11. Writingltheldifferentiallequationlinlthelformlx(dy/dx)l+ly2l =l 1,lwelseelthatlitlislnonlinearlinlylb
ecauselofly2.lHowever,lwritinglitlinlthelforml(y2l—l1)(dx/dy)l+lxl=l0,lwelseelthatlitlisllinearlinlx.
12. Writingltheldifferentiallequationlinlthelformlu(dv/du)l+l(1l+lu)vl =lueulwelseelthatlitlislline
arlinlv.lHowever,lwritinglitlinlthelforml(vl+luvl—
u
lue )(du/dv)l+lul =l 0 ,lwelseelthatlitlislnonlinearlinlu.

13. Fromlyl=le−x/2lwelobtainlyjl=l—l1le2−x/2.lThenl2yjl+lyl=l—e−x/2l+le−x/2l=l0.




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,SolutionlandlAnswerlGuide:lZill,lDIFFERENTIALlEQUATIONSlWithlMODELINGlAPPLICATIONSl2024,l9780357760192;lChapterl#1:
IntroductionltolDifferentiallEquations


6 6 —
14. Fromlyl = — e 20t welobtainldy/dtl =l24e −20t ,lsolthat
5 5
ll
dy −20t l
+l20yl=l24e +l20 6 —l 6e −20t =l 24.
dt 5l 5l


15. Fromlyl=le3xlcosl2xlwelobtainlyjl =l3e3xlcosl2x—2e3xlsinl2xlandlyjjl =l5e3xlcosl2x—
12e3xlsinl2x,lsolthatlyjjl—l6yjl+l13yl=l0.
j
16. Fromlyl =l —lcoslxlln(seclxl+ltanlx)lwelobtainly =l—1l+lsinlxlln(seclxl+ltanlx)land
jj jj
yl =ltanlxl+lcoslxlln(seclxl+ltanlx).lThenlyl +lyl=ltanlx.
17. Theldomainloflthelfunction,lfoundlbylsolvinglx+2 l ≥l 0,lisl[—2,l∞).l Fromlyjl =l 1+2(x+2)−1/2
welhave
j −1/2
(yl — x)yl =l(yl—lx)[1l+l(2(xl+l2) ]

=lyl—lxl+l2(yl— x)(xl+l2) −1/2

=lyl—lxl+l2[xl+l4(xl+l2)1/2l— x](xl+l2) −1/2

=lyl—lxl+l8(xl+l2)1/2 (xl+l2) −1/2l =lyl —lxl+l8.

Anlintervallofldefinitionlforlthelsolutionlofltheldifferentiallequationlisl(—
2,l∞)lbecauselyjlislnotldefinedlatlxl=l—2.
18. Sinceltanlxlislnotldefinedlforlxl =l π/2l+lnπ,lnlanlinteger,ltheldomainloflyl =l 5ltanl5xlis
{xl ll 5xl/=lπ/2l+lnπ}
j lsecl 5x2lwelhave
orl{xl xl/=lπ/10l+lnπ/5}.lFromlyl =l25
j
y =l25(1l+ltan 2 5x)l=l25l+l25ltan 2
5xl=l25l+ly 2 .

Anlintervallofldefinitionlforlthelsolutionlofltheldifferentiallequationlisl(—π/10,lπ/10).lAn-
lotherlintervallis l(π/10,l3π/10),landlsolon.

19. Theldomainloflthelfunctionlisl{x l 4l—lx2 /=l 0}lorl{x l xl /=l — j

2lorlxl /=l 2}.lFromlyl =l2x/(4l—lx2)2lwelhave
l l
2
1
l l l
yjl =l2x =l 2xy2.
4l—lx2
Anlintervallofldefinitionlforlthelsolutionlofltheldifferentiallequationlisl(—2,l2).lOtherlinter-
lvalslarel(—∞,l—2)landl(2,l∞).
√
20. Thelfunctionlislyl =l1/ 1l—lsinlxl,lwhoseldomainlislobtainedlfroml1l—lsinlxl/=l 0lorlsinlxl /=l 1.
Thus,ltheldomainlisl{xl xl/=lπ/2l+l2nπ}.lFromlyl =l— j l (1l—1
lsinlx)
−3/2 (—lcoslx)lwelhave
2

2yjl=l(1l—lsinlx)−3/2lcoslxl=l[(1l—lsinlx)−1/2]3lcoslxl=ly3lcoslx.

Anlintervallofldefinitionlforlthelsolutionlofltheldifferentiallequationlisl(π/2,l5π/2).lAnotherlonelis
l(5π/2,l9π/2),landlsolon.



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, SolutionlandlAnswerlGuide:lZill,lDIFFERENTIALlEQUATIONSlWithlMODELINGlAPPLICATIONSl2024,l9780357760192;lChapterl#1:
IntroductionltolDifferentiallEquations




21. Writinglln(2Xl —l 1)l —l ln(Xl —l 1)l =l tlandldifferentiating x

implicitlylwelobtain 4

2 dX 1 dX
—l =l1 2
2Xl—l1l dtl Xl—l1l dtl
l ll
2 l ll1 dXl –l4 –2 t
—l =l1 2 4
2Xl—l1 Xl—l1 dt
–2
2Xl—l2l—l2Xl+l1ldXl
=l1
(2Xl—l1)l(Xl—l1)l dt –l4
dX
=l—(2Xl—l1)(Xl—l1)l=l(Xl—l1)(1l—l2X).
dtl

Exponentiatinglbothlsidesloflthelimplicitlsolutionlwelobtain

2Xl—
l1l Xl—l1
=letl
2Xl—l1l=lXetl—let

(etl—l1)l=l(etl—l2)X
et — 1
X l =l .
etl —l2l

Solvingletl—l2l=l0lwelgetltl=llnl2.lThus,lthelsolutionlisldefinedlonl(—
∞,llnl2)lorlonl(lnl2,l∞).lThelgraphloflthelsolutionldefinedlonl(—
∞,llnl2)lisldashed,landlthelgraphloflthelsolutionldefinedlonl(lnl2,l∞)lislsolid.

22. Implicitlyl differentiatinglthel solution,l welobtain y

2l dy dy 4
—2xl —l4xyl+l2yl =l0
dxl dxl
2
—x2l dyl—l2xyldxl+lyldyl=l0
x
2xyldxl+l(x2l—ly)dyl=l0. –l4 –2 2 4

–2
Usinglthelquadraticlformulaltolsolvely2l —l 2x2yl —l 1l =l 0
√ √
forly,lwelgetlyl = 2x2 ± 4x4l +l4l /2l =l x2 ± x4l +l1l. –l4
√
Thus,ltwolexplicitlsolutionslarely1l =l x2l+l x4l+l1l and
√
y2l =l x2l —l x4l +l1l.l Bothlsolutionslareldefinedlonl(—∞,l∞).
Thelgraphlofly1(x)lislsolidlandlthelgraphlofly2l isldashed.




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