Student Solutions Manual for Elementary Differential Equations and Elementary Differential Equations with Boundary Value Problems

288 Pages·2016·1.66 MB·English

STUDENT SOLUTIONSMANUAL FOR
ELEMENTARY
DIFFERENTIAL EQUATIONS
AND
ELEMENTARY
DIFFERENTIAL EQUATIONS
WITH BOUNDARY VALUE
PROBLEMS
William F. Trench
Andrew G.CowlesDistinguishedProfessor Emeritus
DepartmentofMathematics
TrinityUniversity
SanAntonio,Texas, USA
[email protected]
©2013WilliamF.Trench,allrightsreserved
ThisbookwaspublishedpreviouslybyBrooks/ColeThomsonLearning
Reproductionispermittedforanyvalidnoncommercialeducational,mathematical,orscientificpurpose.
However,chargesforprofitbeyondreasonableprintingcostsareprohibited.
CORRECTIONSAREWELCOME
TO BEVERLY
Contents
Chapter1 Introduction 1
1.2 FirstOrderEquations 1
Chapter2 FirstOrderEquations 5
2.1 LinearFirstOrder Equations 5
2.2 SeparableEquations 8
2.3 ExistenceandUniquenessofSolutionsofNonlinearEquations 11
2.4 TransformationofNonlinearEquationsintoSeparableEquations 13
2.5 ExactEquations 17
2.6 IntegratingFactors 21
Chapter3 NumericalMethods 25
3.1 Euler’sMethod 25
3.2 TheImprovedEulerMethodandRelatedMethods 29
ii Contents
3.3 TheRunge-KuttaMethod 34
Chapter4 ApplicationsofFirstOrderEquations 39
4.1 GrowthandDecay 39
4.2 CoolingandMixing 40
4.3 ElementaryMechanics 43
4.4 AutonomousSecondOrderEquations 45
4.5 ApplicationstoCurves 46
Chapter5 LinearSecondOrderEquations 51
5.1 HomogeneousLinearEquations 51
5.2 ConstantCoefficientHomogeneousEquations 55
5.3 NonhomgeneousLinearEquations 58
5.4 TheMethodofUndeterminedCoefficientsI 60
5.5 TheMethodofUndeterminedCoefficientsII 64
5.6 ReductionofOrder 75
5.7 VariationofParameters 79
Chapter6 ApplcationsofLinearSecondOrderEquations 85
6.1 SpringProblemsI 85
6.2 SpringProblemsII 87
6.3 TheRLCCircuit 89
6.4 MotionUnderaCentralForce 90
Chapter7 SeriesSolutionsofLinearSecondOrderEquations 108
7.1 ReviewofPowerSeries 91
7.2 SeriesSolutionsNearanOrdinaryPointI 93
7.3 SeriesSolutionsNearanOrdinaryPointII 96
7.4 RegularSingularPoints;EulerEquations 102
7.5 TheMethodofFrobeniusI 103
7.6 TheMethodofFrobeniusII 108
7.7 TheMethodofFrobeniusIII 118
Chapter8 LaplaceTransforms 125
8.1 IntroductiontotheLaplaceTransform 125
8.2 TheInverseLaplaceTransform 127
8.3 SolutionofInitialValueProblems 134
8.4 TheUnitStepFunction 140
8.5 ConstantCoefficientEquationswithPiecewiseContinuousForcing
Functions 143
8.6 Convolution 152
Contents iii
8.7 ConstantCofficientEquationswithImpulses 55
Chapter9 LinearHigherOrderEquations 159
9.1 IntroductiontoLinearHigherOrderEquations 159
9.2 HigherOrderConstantCoefficientHomogeneousEquations 171
9.3 UndeterminedCoefficientsforHigherOrderEquations 175
9.4 VariationofParametersforHigherOrderEquations 181
Chapter10 LinearSystemsofDifferentialEquations 221
10.1 IntroductiontoSystemsofDifferentialEquations 191
10.2 LinearSystemsofDifferentialEquations 192
10.3 BasicTheoryofHomogeneousLinearSystems 193
10.4 ConstantCoefficientHomogeneousSystemsI 194
10.5 ConstantCoefficientHomogeneousSystemsII 201
10.6 ConstantCoefficientHomogeneousSystemsII 245
10.7 VariationofParametersforNonhomogeneousLinearSystems 218
Chapter 221
11.1 EigenvalueProblemsfory (cid:21)y 0 221
00C D
11.2 FourierExpansionsI 223
11.3 FourierExpansionsII 229
Chapter12 FourierSolutionsofPartialDifferentialEquations 239
12.1 TheHeatEquation 239
12.2 TheWaveEquation 247
12.3 Laplace’sEquationinRectangularCoordinates 260
12.4 Laplace’sEquationinPolarCoordinates 270
Chapter13BoundaryValueProblemsforSecondOrderOrdinaryDifferentialEquations 273
13.1Two-PointBoundaryValueProblems 273
13.2Sturm-LiouvilleProblems 279
CHAPTER 1
Introduction
1.2 BASICCONCEPTS
1.2.2. (a)Ify ce2x,theny 2ce2x 2y.
xD2 c 0 D 2x Dc 2x2 c x2 c
(b)Ify ,theny ,soxy y x2.
D 3 C x 0 D 3 (cid:0) x2 0C D 3 (cid:0) x C 3 C x D
(c)If
1
y ce x2; then y 2xce x2
(cid:0) 0 (cid:0)
D 2 C D(cid:0)
and
1
y 2xy 2xce x2 2x ce x2 2xce x2 x 2cxe x2 x:
0 (cid:0) (cid:0) (cid:0) (cid:0)
C D(cid:0) C 2 C D(cid:0) C C D
(cid:18) (cid:19)
(d)If
1 ce x2=2
y C (cid:0)
D 1 ce x2=2
(cid:0) (cid:0)
then
.1 ce x2=2/. cxe x2=2/ .1 ce x2=2/cxe x2=2
y (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) C (cid:0) (cid:0)
0 D .1 cxe x2=2/2
(cid:0) (cid:0)
2cxe x2=2
(cid:0) (cid:0)
D .1 ce x2=2/2
(cid:0) (cid:0)
and
1 ce x2=2 2
y2 1 C (cid:0) 1
(cid:0) D 1(cid:0)ce(cid:0)x2=2! (cid:0)
.1 ce x2=2/2 .1 ce x2=2/2
C (cid:0) (cid:0) (cid:0) (cid:0)
D .1 ce x2=2/2
(cid:0) (cid:0)
4ce x2=2
(cid:0) ;
D .1 ce x2=2/2
(cid:0) (cid:0)
1
2 Chapter1BasicConcepts
so
4cx 4cx
2y x.y2 1/ (cid:0) C 0:
0C (cid:0) D .1 ce x2=2/2 D
(cid:0) (cid:0)
x3 x3 x3
(e)Ify tan c ,theny x2sec2 c x2 1 tan2 x2.1 y2/.
D 3 C 0 D 3 C D C 3 c D C
(f)If (cid:18) (cid:19) y .c1 (cid:18)c2x/ex (cid:19)sinx (cid:18)x2; the(cid:18)n C (cid:19)(cid:19)
D C C C
y .c 2c x/ex cosx 2x;
0 1 2
D C C C
y .c 3c x/ex sinx 2;
0 1 2
D C (cid:0) C
and y 2y y c ex.1 2 1/ c xex.3 4 1/
00 0 1 2
(cid:0) C D (cid:0) C C (cid:0) C
sinx 2cosx sinx 2 4x x2
(cid:0) (cid:0) C C (cid:0) C
2cosx x2 4x 2:
D (cid:0) C (cid:0) C
2 2 4
(g)Ify c ex c x ,theny c ex c andy c ex ,so.1 x/y xy y
D 1 C 2 Cx 0 D 1 C 2(cid:0)x2 00 D 1 Cx3 (cid:0) 00C 0(cid:0) D
4.1 x/ 2 2 4.1 x x2/
c .1 x x 1/ c .x x/ (cid:0) (cid:0) (cid:0)
1 (cid:0) C (cid:0) C 2 (cid:0) C x3 (cid:0) x (cid:0) x D x3
c sinx c cosx c cosx c sinx c sinx c cosx
1 2 1 2 1 2
(h)Ify C 4x 8theny (cid:0) C 4and
D x1=2 C C 0 D x1=2 (cid:0) 2x3=2 C
c sinx c cosx c sinx c cosx 3c sinx c cosx 1
y 1 C 2 1 (cid:0) 2 1 C 2 ,sox2y xy x2 y
00 D(cid:0) x1=2 (cid:0) x3=2 C4 x5=2 00C 0C (cid:0) 4 D
(cid:18) (cid:19)
3
c x 3=2sinx x1=2cosx x 1=2sinx x1=2cosx
1 (cid:0) (cid:0)
(cid:0) (cid:0) C 4 C (cid:0)
(cid:18)
1 1 3
x 1=2sinx x3=2sinx x 1=2sinx c x 3=2cosx x1=2sinx x 1=2cosx
(cid:0) (cid:0) 2 (cid:0) (cid:0)
2 C (cid:0) 4 C (cid:0) C C 4
(cid:19) (cid:18)
1 1 1
x1=2sinx x 1=2cosx x3=2cosx x 1=2cosx 4x x2 .4x 8/ 4x3 8x2
(cid:0) (cid:0)
(cid:0) (cid:0) 2 C (cid:0) 4 C C (cid:0) 4 C D C C
(cid:19) (cid:18) (cid:19)
3x 2.
(cid:0)
1.2.4. (a)Ify xex,theny xex exdx c .1 x/ex c,andy.0/ 1 1 1 c,
0 D(cid:0) D(cid:0) C C D (cid:0) C D ) D C
soc 0andy .1 x/ex.
D D (cid:0) R
1 (cid:25)
(b) If y xsinx2, then y cosx2 c; y 1 1 0 c, so c 1 and
0 D D (cid:0)2 C 2 D ) D C D
(cid:18)r (cid:19)
1
y 1 cosx2.
D (cid:0) 2
sinx 1 d
(c) Write y tanx .cosx/. Integrating thisyields y ln cosx c;
0 D D cosx D (cid:0)cosxdx D (cid:0) j jC
y.(cid:25)=4/ 3 3 ln.cos.(cid:25)=4// c,or3 lnp2 c,soc 3 lnp2,soy ln. cosx /
D ) D(cid:0) C D C D (cid:0) D(cid:0) j j C
3 lnp2 3 ln.p2 cosx /.
(cid:0) D (cid:0) j j x5 32 37
(d) If y x4, then y c ; y .2/ 1 c 1 c , so y
00 D 0 D 5 C 1 0 D (cid:0) ) 5 C 1 D (cid:0) ) 1 D (cid:0)15 0 D
x5 37 x6 37 64 47
. Therefore, y .x 2/ c ; y.2/ 1 c 1 c , so
2 2 2
5 (cid:0) 15 D 30 (cid:0) 15 (cid:0) C D (cid:0) ) 30 C D (cid:0) ) D (cid:0)15
47 37 x6
y .x 2/ .
D(cid:0)15 (cid:0) 5 (cid:0) C 30
xe2x 1 xe2x e2x xe2x e2x
(e) (A) xe2xdx e2xdx . Therefore, y c ;
D 2 (cid:0) 2 D 2 (cid:0) 4 0 D 2 (cid:0) 4 C 1
R 1 5 Z 5 xe2x e2x 5 xe2x
y .0/ 1 c c , so y ; Using(A)again, y
0 D ) (cid:0)4 C 1 D 4 ) 1 D 4 0 D 2 (cid:0) 4 C 4 D 4 (cid:0)
e2x e2x 5 xe2x e2x 5 1 29
x c x c ; y.0/ 7 c 7 c , so
2 2 2 2
8 (cid:0) 8 C 4 C D 4 (cid:0) 4 C 4 C D ) (cid:0)4 C D ) D 4
xe2x e2x 5 29
y x .
D 4 (cid:0) 4 C 4 C 4
(f)(A) xsinxdx xcosx cosxdx xcosx sinx and(B) xcosxdx xsinx
D (cid:0) C D (cid:0) C D (cid:0)
sinxdx xsinx cosx. Ify xsinx,then(A)impliesthaty xcosx sinx c ;y .0/
RD C 00 DR(cid:0) 0 D R (cid:0) C 1 0 D
3 c 3,soy xcosx sinx 3. Now(B)impliesthaty xsinx cosx cosx 3x c
(cid:0)R ) D(cid:0) 0 D (cid:0) (cid:0) D C C (cid:0) C 2 D
xsinx 2cosx 3x c ;y.0/ 1 2 c 1 c 1,soy xsinx 2cosx 3x 1.
2 2 2
C (cid:0) C D ) C D ) D(cid:0) D C (cid:0) (cid:0)
Section1.2BasicConcepts 3
(g) If y x2ex, then y x2exdx x2ex 2 xexdx x2ex 2xex 2ex c ;
000 D 00 D D (cid:0) D (cid:0) C C 1
y .0/ 3 2 c 3 c 1,so(A)y .x2 2x 2/ex 1. Since .x2 2x 2/exdx
00 D ) C 1 D ) 1 D R 00 D (cid:0) CR C (cid:0) C D
.x2 2x 2/ex .2x 2/exdx .x2 2x 2/ex .2x 2/ex 2ex .x2 4x 6/ex,
(cid:0) C (cid:0) (cid:0) D (cid:0) C (cid:0) (cid:0) C R D (cid:0) C
(A)impliesthaty .x2 4x 6/ex x c ; y .0/ 2 6 c 2 c 8, so(B)
0 DR (cid:0) C C C 2 0 D (cid:0) ) C 2 D (cid:0) ) 2 D (cid:0)
y .x2 4x 6/ex x 8;Since .x2 4x 6/exdx .x2 4x 6/ex .2x 4/exdx .x2
0 D (cid:0) C C (cid:0) (cid:0) C D (cid:0) C (cid:0) (cid:0) D (cid:0)
x2
4x 6/ex .2x 4/ex 2ex .x2 R6x 12/ex,(B)impliesthaty .x2 6xR 12/ex 8x c ;
3
C (cid:0) (cid:0) C D (cid:0) C D (cid:0) C C 2 (cid:0) C
x2
y.0/ 1 12 c 1 c 11,soy .x2 6x 12/ex 8x 11.
3 3
D ) C D ) D(cid:0) D (cid:0) C C 2 (cid:0) (cid:0)
cos2x 1 7
(h) Ify 2 sin2x, then y 2x c ; y .0/ 3 c 3 c ,
000 D C 00 D (cid:0) 2 C 1 00 D ) (cid:0)2 C 1 D ) 1 D 2
cos2x 7 sin2x 7
so y 2x . Then y x2 x c ; y .0/ 6 c 6, so
00 D (cid:0) 2 C 2 0 D (cid:0) 4 C 2 C 2 0 D (cid:0) ) 2 D (cid:0)
sin2x 7 x3 cos2x 7 1 7
y x2 x 6. Theny x2 6x c ;y.0/ 1 c 1 c ,
0 D (cid:0) 4 C2 (cid:0) D 3 C 8 C4 (cid:0) C 3 D ) 8C 3 D ) 3 D 8
x3 cos2x 7 7
soy x2 6x .
D 3 C 8 C 4 (cid:0) C 8
(i)Ify 2x 1,theny x2 x c ;y .2/ 7 6 c 7 c 1;soy x2 x 1.
000 Dx3 Cx2 00 D C C 1 00 D14 ) C 1D ) 1 D26 00 Dx3C xC2
Theny .x 2/ c ;y .2/ 4 c 4 c ,soy
0 D 3 C 2 C (cid:0) C 2 0 D(cid:0) ) 3 C 2D(cid:0) ) 2 D(cid:0) 3 0 D 3 C 2 C
26 x4 x3 1 26 8 5
.x 2/ .Theny .x 2/2 .x 2/ c ;y.2/ 1 c 1 c ,
3 3 3
(cid:0) (cid:0) 3 D 12C 6 C2 (cid:0) (cid:0) 3 (cid:0) C D ) 3C D ) D(cid:0)3
x4 x3 1 26 5
soy .x 2/2 .x 2/ .
D 12 C 6 C 2 (cid:0) (cid:0) 3 (cid:0) (cid:0) 3
1.2.6. (a)Ify x2.1 lnx/,theny.e/ e2.1 lne/ 2e2;y 2x.1 lnx/ x 3x 2xlnx,
D C D C D 0 D C C D C
so y .e/ 3e 2elne 5e; (A)y 3 2 2lnx 5 2lnx. Now, 3xy 4y 3x.3x
0 D C D 00 D C C D C 0 (cid:0) D C
2xlnx/ 4x2.1 lnx/ 5x2 2x2lnx x2y ,from(A).
(cid:0) x2C D C 1 D 00 1 2 2 5
(b)Ify x 1,theny.1/ 1 1 ; y x 1, soy .1/ 1 ; (A)
D 3 C (cid:0) D 3 C (cid:0) D 3 0 D 3 C 0 D 3 C D 3
2 2 x2 2
y . Nowx2 xy y 1 x2 x x 1 x 1 1 x2 x2y ,from(A).
00 D 3 (cid:0) 0C C D (cid:0) 3 C C 3 C (cid:0) C D 3 D 00
(cid:18) (cid:19)
(c)Ify .1 x2/ 1=2, theny.0/ .1 02/ 1=2 1;y x.1 x2/ 3=2, soy .0/ 0;(A)
D C (cid:0) D C (cid:0) D 0 D (cid:0) C (cid:0) 0 D
y .2x2 1/.1 x2/ 5=2.Now,.x2 1/y x.x2 1/y .x2 1/.1 x2/ 1=2 x.x2 1/. x/.1
00 D (cid:0) C (cid:0) (cid:0) (cid:0) C 0 D (cid:0) .xC2 1(cid:0)/y (cid:0)x.x2 C1/y(cid:0) C
x2/ 3=2 .2x2 1/.1 x2/ 1=2 y .1 x2/2from(A),soy (cid:0) (cid:0) C 0.
(cid:0) D (cid:0) C (cid:0) D 00 C 00 D .x2 1/2
x2 1=4 1 x.x 2/ C. 1=2/. 3=2/
(d)Ify ,theny.1=2/ ;y (cid:0) ,soy .1=2/ (cid:0) (cid:0) 3;
D 1 x D 1 1=2 D 2 0 D(cid:0).1 x/2 0 D .1 1=2/2 D
2(cid:0) (cid:0) x2 x (cid:0) x2.x (cid:0)2/ x2
(A)y . Now,(B)x y x and(C)xy y (cid:0)
00 D .1 x/3 C D C1 x D 1 x 0(cid:0) D(cid:0) .1 x/2 (cid:0)1 x D
x2 (cid:0) (cid:0) x3(cid:0) x3 2.x (cid:0)y/.xy y(cid:0)/
. From(B)and(C),.x y/.xy y/ y ,soy C 0 (cid:0) .
.1 x/2 C 0 (cid:0) D .1 x/3 D 2 00 00 D x3
(cid:0) (cid:0)
1.2.8. (a) y .x c/a is defined and x c y1=a on .c; /; moreover, y a.x c/a 1
a y1=a a(cid:0)1 DDay.a(cid:0)(cid:0)1/=a. (cid:0) D 1 0 D (cid:0) (cid:0) D
(b)ifa>1ora <0,theny 0isasolutionof(B)on. ; /.
(cid:0) (cid:1) (cid:17) (cid:0)1 1
1.2.10. (a)Sincey c wemustshowthattherightsideof(B)reducestoc forallvaluesofx insome
0
D
4 Chapter1BasicConcepts
interval.Ify c2 cx 2c 1,
D C C C
x2 4x 4y x2 4x 4c2 4cx 8c 4
C C D C C C C C
x2 4.1 c/x 4.c2 2c 1/
D C C C C C
x2 4.1 c/ 2.c 1/2 .x 2c 2/2:
D C C C C D C C
Therefore, x2 4x 4y x 2c 2andtherightsideof(B)reducestocifx > 2c 2.
C C D C C (cid:0) (cid:0)
x.x 4/ x 2
(b)Ify1 Dp(cid:0) 4C ,theny10 D(cid:0) C2 andx2C4xC4y D0forallx. Therefore, y1 satisfies
(A)on. ; /.
(cid:0)1 1
CHAPTER 2
First Order Equations
2.1 LINEARFIRSTORDEREQUATIONS
y
2.1.2. 0 3x2; ln y x3 k; y ce x3. y ce .lnx/2=2.
y D(cid:0) j j jD(cid:0) C D (cid:0) D (cid:0)
y 3 c
2.1.4. 0 ; ln y 3ln x k ln x 3 k; y .
y D(cid:0)x j jD(cid:0) j jC D(cid:0) j j C D x3
y 1 x 1 ce x
2.1.6. 0 C 1; ln y ln x x k; y (cid:0) ; y.1/ 1 c e;
y D (cid:0) x D (cid:0)x (cid:0) j j j D (cid:0) j j(cid:0) C D x D ) D
e .x 1/
y (cid:0) (cid:0) .
D x
y 1 c
2.1.8. 0 cotx; ln y ln x ln sinx k ln xsinx k; y ;
y D (cid:0)x (cid:0) j j j D (cid:0) j j(cid:0) j jC D (cid:0) j jC D xsinx
(cid:25)
y.(cid:25)=2/ 2 c (cid:25); y .
D ) D D xsinx
y k
2.1.10. 0 ; ln y kln x k ln x k k ; y c x k; y.1/ 3 c 3;
y D (cid:0)x j j j D (cid:0) j jC 1 D j (cid:0) jC 1 D j j(cid:0) D ) D
y 3x k.
(cid:0)
D
y
2.1.12. 10 3; ln y 3x; y e 3x; y ue 3x; ue 3x 1; u e3x; u
y D (cid:0) j 1j D (cid:0) 1 D (cid:0) D (cid:0) 0 (cid:0) D 0 D D
1
e3x 1
c; y ce 3x.
(cid:0)
3 C D 3 C
y
2.1.14. 10 2x; ln y x2; y e x2; y ue x2; ue x2 xe x2; u x;
y D (cid:0) j 1j D (cid:0) 1 D (cid:0) D (cid:0) 0 (cid:0) D (cid:0) 0 D
1
x2 x2
u c; y e x2 c .
D 2 C D (cid:0) 2 C
(cid:18) (cid:19)
y 1 1 u u 7 7
2.1.16. 10 ; ln y ln x ; y ; y ; 0 3; u 3x;
y D (cid:0)x j 1j D (cid:0) j j 1 D x D x x D x2 C 0 D x C
1
3x2 7ln x 3x c
u 7ln x c; y j j .
D j jC 2 C D x C 2 C x
5