Advanced Engineering Mathematics Solutions Manual

Chapter 1

Section 1.2

Section 1.2

1

3.

(a.)

s~

4W=w-4x=y4s)

~~

14!

M=JA-tk>/&x

=;v/l+tj'?...

~(Ila.)

tlNJ.

cub),

fC(.)&

=

A

1

/:l-0

T~e:: ~6=~

1

=&=-t'~1+~

1

i.

u

+

.a

'r~~=f-r~1~i.l#~+B

TC<.>e

A

A

Ov-t\d.

&/~ ~

'()

=

c

11+~,z,

.

.

Chapter 2

I

Section

2.2

1

4 ..

1

Section 2.3

1

5

Section 2.3

- sx

R cl'C

t

s;

E«)l

2..

(b)

,i.(;1:)

=

e

0 L

(Jo

e

E;:)

d.t+Lo)

t<t

₁ '(f~

El-c)=o

iiv..

~'\.t IV"""'(Y--..J MJ .Mt)=

e

(o+J..

₀ =J.₀

e

.

4

tJ t.L .~1): .. n -O • -R:t/L. • ) ,

-Rt/L

.

i">1:1_~(

t R't/L . \ E

-fct-:t,)

,

-Rt/L

A.lt)::

e

ti..

Ix,

e

to

i:l'C Ho)=

t'(l-e

)

Hoe

~

S~AtSt:

.let)+ ErR \

-- -.. - ----. ---.. -. --- -- --'*'- ----·

.

Jwo

..

Section 2.3 ·

1

6

~

R~(/Jt-<i>)

=

A~otCf.JW<t>-~ct>cr.>tJt). ~~

<~~~d

~*)

A~<P='-'L1 'D~

~

4::wL/r<

dl

4>=~

1

[wL/£)) ~

.

ACf.>cl>

=

!5

~"i

~

~

~

N-=

~+(IJL)t..

.40

A=

4Rz.H<.JL)'l..

rp~

i.Ct)=: t;,iJL eRt/IQ

+

~ ~(tJ.t-<f>).

)

R2°+(tJLY°

~

R

2

₊

_(t.JL)t

6.

m

f :t)

=

IY\o

ekt

/:>() g::

toe-{,()~ ~

-

w-k :: ;,.,. o.g.)

1.

=

0.00312..

,60 rn(t)

=

toe-

0 ·003'72.! . 2.:: 10

e ·

00312;t

~

.t

=

432.C,

~J

aJWl

0.1::: 10

e

0·003'12

t

~

;t

=

12.37.9 MJ10f.

-Jc.;t

rr./ J -70~ " - - ,_ 0 ~~

'7.

m.(t)

=

m

₀

e

.

o.Sr-

₀

=

'floe

~ .If{_= 0.003188. ll\M.\.)

(). 5'JP{o

=

o/.,

e-o.003Y-88T

~

rp

~ 21~.4 ~:

12 ••

(ll)

Ynt\f

1

=rrt~-Cl\rj

(\j(o)=.0.

~

N"

1

+~M=~ (~-d\Jut~~·)Ao

Ntt)

=

e -

.r

c.tt/111. (

s

ef

cctt/m

~

tlt" t

A )

=

e

-Ct/M (

~

s

e

d:/m

d.:t

+

fll )

::: rM.

+

Aed/M..

rr~

rv-to)

=o

=~+A ~

A=-

~c ~cl

-CO C 0 « , ft _ tl - ,,

N"(t)

=

~(1-e-ct/m

).

a.o

t~c0> l\rlt)~-% =~~ ~~.

(b)

ml\T'=

l'\'ld-c""z.

M>

~

"'-

R~~: :ii~-

(flu.

~~II"""

S.u..2:2.J

With.

X.J

'd

~

10

t,

l\f,

~

f>Ct) - -

<;Im 1

jf

:t)

=O,

~:t)

=

ef ·

(9'~J.

tktp~~

~m~c) ~

~~~

~j.Jo

t\f{t)=~m~/c+d£rj. ~

ih<.

ODE~

o-~=d- ~O~+_t:)

=

f

~(~u-l~-ix +~)

r:J"L

u.'-2~~u.== ~ ~

~

M.Ct)-

_J_

~

+Ae1.1~C/rl\';t rp~ rV{O)=D=1%·+_j_ ~M{O)=-~c

- 2.

'i

~

·

J O Ll(O) d - ~

==-±.[;:~+A ~

A=-1~~. F~J f\fCtJ::J~

+iilt)

=~~

+

-t~ -f~ e~1¥'£x

=

pj(1-

J+~2~x) ~<!4

~ ~

~&~~""'~·

13. This problem is worked in the Answers to Selected Exercises. Here, we just wish to mention that to help the student feel more comfortable. about the physical process of light extinction it might be useful to note the gradual extinction of light as we proceed deeper and deeper into the ocean.

14. NOTE: This problem is nice for use in class or lecture, especially in view of its environmental interest. Later on it will also make a nice example for the application of the Fourier transform, especially if the source is modeled as Q times a delta function at x = 0. The solution is given in the Answers to Selected Exercises, so here we will just give sketches of the results and (for possible class discussion) give a brief formal derivation of the

governing ODE.

(CL) C (b)

_c

Section 2.4

7

'T'o

~

~

~\r''"'',.,;.~ ODE~ ~.ii. Yv)~ ~

~

IN>\

~

~

i

i1't

.n.wtri,

~

x

~

~+Ll:i'.

F.A.Ck>,o,

k i

~

~

-tt-..:t

~

fkw- {

/'IY\M(). c<>1

~)

~

&-e

"~,,

1

AnUI,.

A

u-

~

...o

N.~oJ.

:to.

J4

~f\'b~<fht.~~

-c'cx)

(~~,,,X4,~MtJ.e

k~¥

~~

:tOk

~,

r.>o

.c'cx>>O,w.J.Q~ ':\-~J

~

~)

.io-t4

~.Jt ~.. c'(~)

<

0

.wJ.l

~

~ f~

JO~ n~,:

l-r)

~

~

ctv:iil.£4.

~~.-#?. ~-'°'CA.~~~~~

~.

~l\rUl. ~ ~

.4t

<2t4,

~

1

f~

't..

o..vJ.

...xi.~ ~

~k

~~:

~:!>~: €cc~>A1::,X,~t

D*-:

-~Ac'c~>.at

D~~.: --k.Ac'cx.+~-x,)llt

~: (llit)Acc~) ~: (U6t)Ac.c,c,+~)

~ x+6x,

1)~~~:

Qc~)AXAt

..

..

~

tk

h

~Jo ~.J_ ~

M,

€.

t.ui-~MMD ~

.i-.t

~

AN-A.

Qc4X)

>.(,

~

~

~

Jk

~

~

~

~k~

'ftJ\

MMl-

~.

NdWJ

n~

~

JW\fM>O,

°",#

~

~

~

.

. ....

~

b,~ ~

'rM..ll.

:t.-.i

.t.t

=

ma.oo

AN'\ - IY\<lllO "'"')

AO E.c("/.,)

Allx.t:>t

=

[-JcAc..'

(X.).At

-(U.6t)

Ptccix.)]

. .

-

[-J:f\c'c~+6X.)~t-(Ua.) Acc~+6X.)+

Qc?<'..)6ut

J

J)~

~

ALJ<X$.

~

~

4X:~O ~

-kc'' -

Uc

_JIDc

= -

~

Let

..ua

eaM.

~

f

15.

(0..)

~

+-k.t-l

=

.m

~

A.I.Ct)=

e

S-Mt (

s

eS-hlt

klictt

+

c)

=

U

+

c e.kt.

M.{0)=.4₀

=

U+C

~

C=-40-U,

M:J M(.t)=

Lto+

U(1-e---kt).

''·(Cl)

Set>=

S

₀

(1+

~)"t

=

S

0

(1+~)fi)J:.t

==

So(1+f.n,,ttt

~ S

0

e~t DJ.:>m~oo.

Section 2.4

8

Section 2.4

10

Cf) /\.\

'=

~+2~-1

.

L4:

x.+2,4-\

=

~

40Jrv=1+2

~

=

l+

2. rr-1 .

~,

~

= .

41'V'--3

0 .2X.+4~-I U (J;l ~ 21\T"-I OIX... "21\r-I

n~:~

cllll"

=

J

~

40

~

rv

+

.g~cs/\)""-')

=

x.+

C, "",

4(X.+z.At

>

+

.k(ax.+1c.11H,

)=

'i!~+A

~i1u.~M....~~·

()

d ..L.

12.

dtJ/&.t

=KN

f,

N-r

d.N

=

k~, ~~-r

= Kt+

c

Cf'*'),

Net>=

[Ct-p)k'.t+A]

i-p.

- L

Fett

f

< 1,

Net) "" gi-r

)l<:t

tr

=

o{t

p

.wk.e.

e

=

1

~r ~

f

~a;.. ff':~

Fo>t

f>>l,

N(t)

=

[A

I

]t-r

-+ 00

a.o.

:t~

A

J

~

AC4.M.k-01~

- Cp-1)1<

:t -

Cp-t)I<

-'-~

~

~

N

₀

~

N

₀

=

A'-r

~

A=

N~-r. 1'~,

NCt)....,.o0

~

.t..+'P,

~

'f

=

1/[q~-t)KNf-

1

_].

_ _

13.

<J..tJ/tU-

=

(a..-bN)N,

N<o)=

N

0 •

W,.;th

t=cd:

~a N

=

JoN/a.J

taN_(a..-b~-N)~t:t

o'l,

m=c1-N)N·

~N'o)=}JoO'lNCo)=bNo:~

-teat -

b b ,.._.

cit

;

o "'

~

'

cUJ_=clt

~N-4CN-t)=t+A

N

=Cet

Nro)=8~C,=~·

N(t-N) J ) N-f ) \ 4

f-(

rn

-

c

~

J2.

t

-

·

1

-kv.o,,

Ntt) =. -

e

= _

@-1

e

N 1-ce~

1-p.

e±

=

(3

~+ (1-~)e-t

14.

L.&:

r,

i,T

~.!.fort

-fcri-ct,

1,,.~

:t

l>vv\tl

~.

~

~)D

.. 2'.r\.a

-~ l'Y\~ ~M-dt ~~~:tr\~:~=:

fi_2.

NOVJ,

v~

DMN\~ P~

<r>~~

L

t

~

rn

~'f1-/'£

x

'£

c j:"')'

/£

*

~/'it

F

~

"'

1/'t'

~o

£.

x~

'<f.IT'

'Po.~~~

t'

~

"'°4

a.

~~

1

tk

~~

-e...o

~

1

T)

~

~ v~) Xo~~~

!.. YV\/C) d1..

c/-k._;

~

~

~

~

o&M.t..

Let

M.()J MM,

l/t..)

J ~ •

ak:t

..\I:)) ;t : ~

=

iAJt .

%

~~~

~

Ml(_ Mo.cl

°'-

~~

"\

ik

'P~_iW

.fuu..,

~

1

;l)

~

a,o, ?lo )

X,~/

CJ)

FI*)

llM.A.

..00 m. .

Let

.u.o M4.t..

?to>~

:

x

=

f .

N~~ ~

ctt

=

ts

ii' ,

d'k

ODE

~

0

h'l.

~

_!

tft

'X.

0

X.lf)

+

C ..1.

G\. _

?G0X.(i )+

ix.

0

£Cf)::.

F~t; X-

0

~(0)=X.0J

CJ CA:> welt"

..L

?G

i

(o)

=

')l'

J.. - 0 0

Section 2.5

11

Section 2.5

13 ·

Cb)

'Tiw.o.,

e-3X#.-,,J2"~

=o.

"W~~~ ~Fl~x=e

3

x~f==Jii

3

x~x=..1c.

~

~F ~

=

-i2.

t:>0 ••• ..1<-,

W

..:t

-'<>

~

{ "-".

~a.U)

;IQ

~

~:

Je""3"-J,x,-

f

'f:z.~

=O'

~

+

~

=

C

,o'l.,

'J(lC.)::

1/(C.+}e

-3X-).

<c)

ct.Ill'..#.-

e21~

=01

s

ep-xiWJA<M-x. -

I

e21tl6d

=

~., ~<~x.>

+te2....,=

c.

&r-,

~<x)::-f~[A-2~(~~>] (2.C-.A,-fd\.~)

9.

(b)

(2n~e+1)cMt. +(l.-z.Cf->~&.&=o, M

9

=~.szCf.)e

=NJt

~

~-Mc.n,e) Noi,e)

df/;}Jt

=

2.n~e-+

1

~

Foz)7)

=

J~'L~e+l)C)5L

=

n-z..µMe+.n.+

A<e-)

'df/"i>e

=

.5l..,_~e-

=

n~e+

A'ce-)

~

Ace)=~.)

./)-0

Fc.n,e)=~ ~

~

~ Jtt.~e+n=

C

(~~fat sue)at·em))1~>·

(c) (2.~-e~)#.+~(~-e1)~ =O) M_'j=2.x.-e~=Nx) .Ao.;V)(~.

dF/ax

=

2x~-e~-+ fC')G,~)

=

S

(2x"j-e'Y) e>~

=

x.'2.'j-Xe~

+

Af"tJ)

cf/t)'d

=

x}·-x.e'd

=

~'1-- ?C.e~

+

A't71)

~

At~)=

cM.

J A:J

'X.~-?le~

=

C.

10.

cr=i.(o'l~ ~~)

11.

(b)

N.t

~·

fdl.

~.

{

Mc,;,~)=

eV3

~

Mc1,1e>==

e"J",

'flu,,,.

M~C')(,?).:

11.ex;}

~

Mx{>J>~)

=

~e.~~

:::/:

~e-x~.

12.

Fc4,b)=-c ..

AO p~ ~ ~ F~~,~)::Fr~,b).

CHAPTER3

Section 3.2

L

(b)

a.

Att

w

LD

~it~~ LI)

.40

~ ~

k

~.

NO.

~.

(b)

1

x~ x.z.+x.)

x_z.4Jx.+1J x.-1}.

c~-z.+~)-(x,z.)::

x

Section 3.2

14

=

t

[(x-i.+~+1)-(x.i.)+ (1--t )]

i..e.)

1 (X'2.+ X.)

-f

(~t.+

x+

l)-

~(Xi.)-

t

{X-1)

=

0.

(~) '1(0) +O(x) + 0[X3)

=

0)

~

,,o,o

A1'.t,

4

ail

~­

(~) '(x.)-3(2.x.)+ocx~)=o, ~

'1,-3,o

o.n.t.4~~-3.

Cb)~ 1'~ 3.2..2:

~·~ eG..J.~

·..

ea.."~

W[e

4

'~

...

leo."x]=

a.,_e"'x

~.e~'X.,···

a..,,,_er...,,:x..

s'dm'l:>7~~10.'t)ih.O

: : :

n-1 a.,>< rH Q.i,X n-1 tln 'X

a.

1

e

a.i

e ···

a.rt

e

I l · · · l

=

rt·~ ea..2_~ •.•

ea."''X,

a.,

a.2. ...

O.n •

rri.t.

.4tt.n.

~~ ~ Q.

v~~~~c~

n-l TH n-1 · . ·

.

.

a.

1

a..

1 • ··

a.n

E~~

1'1 >

~

to.~)

AJ

1

~

a.a't)tll"l(.

~~~~~)~

~v

(~&ea

~~~~~x)

~~·

~~~rr~s.2.2

tkt

1

&-t

a.;~

Ant:

~

~

[

~· ~

... ,

e"·

ic:

1

.i.o.

LI..

s~

>{.

<l'k

a.~'.-o

Aru.

~

~

~<l1't Mt~

L'D. Fd1.

~

a,=~

3

)-fth ~~.

'T'~ 4ea.,~

+oe.o..1.x.

-lf-<!-3""

+oe~x.

+ ... +oeo.._x =o

~

'"11...t.

~~

4,o,-c+,o, ...

,o

~

ill

o.

(c) l HX l+X'Z.

w

[I)\+

x.) \-\-

x.

i..1

=

0 l 2.X

=

~

::

2 =f. 0 ~ ( rp~ 3. 2..

'2.)

LI.

0 0 2.

\.T - ... • • ) ~~ cr.>X ~~ + .... A • J_ ·,.

Ce) \N~x,)c.r,:,x,1~x.

=

cr.>x -~x. ~x, =~= -2PV"\-t'\.?l> ~

-~'X. -~'X. /)vJvX,

A-0~ ~

o

O'Y\.

"'1~.11~ ('1'~3.2..2.),

LI..

cf)

t.

I

x

'X.}-1

'2. - • • -

~

'iJ[?Ci'X.

J=

I 2.X.

=

X-_,~ ~ ~

~

0

cn""'d

~.

t\~ C'.11~3.2.2)J

LI.

s~

~

aJ'\l_

~

t"..o

~

~Jh. ~)

j~~;o

~rr~3.2.?f: ~~a.~~

₁

'ii.t.~;

~)~~LI.

C~)

LI.1ra

'P~3.2.lt.

4.

Ch)

l

~2.X.

Cp2.X,

I

Section 3.5

17

Section 3.6

20

~

.

t.(b)

~=

x,

~

A-1

=o

M ...\=I)

/)=Ax,)

'dl2)=

s-= 2-A

~

A=s/.z

~

~<~)=SX.12.. <-o0<~<oe

cc;

'A'Z..-~+>..

:::-o,

>..=o,o)

(jtx)

=(A+B~~)x

0

= A+B~txJ

=r

A+B~"

-fdi

o< x.<o0

l

A+

Btnc-x.)

f01.

-o0 <

x<

o

3 -J

Ce)

A'l:..A..-rA.-

~

=

o,

A=

±3,

~=Ax

-1: Bx..

()f2>=

1

=

sA+

fYg

~

~

1

r2)=2=

12A-3B/1"

~ 'OC~)

=Zs

x.,3-

f

x.-3 O"n O< X< cO ,

(f)

A'l..-~+i\+

I

=o

₁

A.=

±L

J

~

=

A~(~x.)+8/)w\(~x).

"(1(1)= I=

A)

'j

1(1)=0::

rs )

/)JJ ~(X)

=

C(.)(4x) (SY\ O<X< 0().

(~)

A=2>-l,

~

=Ax'-+R/x.

~t-~)=3::2sA-Bls) ~'f-s)::o:-10A-Bhs; A=1/2s,B~-to)

~ ~l'X..)::: ~2/2..S - t07K. (S'n. - o0 < X< 0

C1r1.)

>..r>--1X>.-2)-2A=oJ

>-.~O,D,3j ~<JI>;;

A+Bk,1"1+ Gx.

3.

0

r1)=

2.=

A+C,

'd'rn=o=l3+3C.

'Cl:m=o

=

-B+c;c,

~

A=2,

13=

c

=o,

-',:1lic.)= 2 ""

-PO,::

x«>o.

cO?

X-A+

X-K2.=o,

~

=

:tK,

1J<x>=

~K

+

B1-if-K

={A

xK+

13~

K O'Y\.

o<

?G< o0

.

A~K+ 8(-~JK

oY\-oO<X.<0

Section 3.6

21

(~)

AJA.--1)(>.-2.)

+

2i\.-2

=o, ,\:::

•11±.i..

j

~(~)=Ax.+

x.[Bep(

1...1?C.1

)+

c~

(!., l'X.I )]

(JV\. 0<' ~ < oO <:fl 6'V\. - oO <

x

< 0.

2.. {m.)

~

(

f

X"2>J

~(~CX),~,?<.-,X,)-2.~~(~C~),x,)

=

0)

~(1)=2) J)(~)'l)=

0

J

:i)(J)(-'j)X1)::0}, "jC'X-)_) j .

r'fk

~·

!)<"-)::

2.

Nott.

if.c.

'D<~X1)

~

J)(J:X~))(a) ~~

fdt_~'(I) ~

~ 'l•)..

.

"·

~

-M

~~~LI

t

tLwt.

~(J{ ~

A.o

~~ ~~

~ik ~.

'P~,~ ~

~..;y(33) ~LI 1~ ~

1

~=

f'Yc·x.)z~ft.i:,Cx.)M ~

-:t

~...!.VJ~'

cl

fYcx)ze-Ja,~tl?<.

=

efa,~/y2rx)=O

A.o-

~

(

Aw-e<.

Ji.e..

-"1f~ ~

~ ~

o

"""'J..

Y'<l<,) ::t:o0)

.o-0~=F~~~~ ~~LI.

7.

(a...)

'X,;t.."a''-xltj'

-3'j

=

O.

'X.=

e\

dlx/tl:t=

ex-,

d:t/d«.=

e-:t

~tlvx.= ,J.Y;~

dt/.14

=

ex

tJ.Y/AX

~~6'x:t.= ~(etg)~

=

(-exrJ:t+e:c~)e-r

,

. M>

ezj;

(-et

~

+

et:~)

er -

e!-

(er~

) -

3Y

=

o,

d.ZY/JJ:.t.. -

2.d.Y/M.

-3Y

=

o,

Yrt)=

Aet+Be

3

~

B.Mt

x.=er~t= ~K,

.40

~(X)

::

Ae""'"

+

B

e

3.k?l

=

Atx,-+

Bx,' .

8.

ca.)

w~ ~

~

7to..)

~

xD~=

T>Y.

~

x.DCxD~)=

J:fY

~

x-z.D'2.'d

+

~

=

D2.Y

O't

~

2

D-z-'d

=

Jf-Y-D'/

=

DlD-1)

'I.

DY

N>U:,

xD (

x2.D-i."J)

=

D

D(D-1)Y 'X3

D

3

-;j

+

2~

=

'J>'l.(D-1)Y

D(D-1)'/

~

-x,

3

_])1~

₌

Jf(D-1)Y-2D(D-1)'/

:: DlD-1)(:D-2)Y)

~AO 6Y\,.

9.cI->)

~=A+B~JL

1

~'CJl1)=0= Bin,~ B=o~ ~t.n)=-A:1iA.v\.

p{JiJ=4'i.=A)

~ ~fn>=q5

2

10. Cb) J..l.

=A+

B/Jt

.U.1lJ1,)

=

3

=

-B/5l~

9

B

=

-3J2.-:- /)0 ,U{Jt)=

A-

3ll~/~. ~

..U.(Jl2)==0

=

A-

3.ll.f

/Jl.z

~

A=

3Jlfl.R.z. /llO M(Jt):: g;p_f

(f--:})

11.cb)

s...Jc.

o<~>=

At'Jl)x.

-a'=A+A'x.,

~"=A'+A'+A"'X-

AO 2.

xC2A'+A''x)+

~l~A'~)-/}h=o) ~2.A'

1

+(2'X.+X.7..)A

1

=

o

o1t,

~

A'=P',

~

ft 11

B

A'

B-~-2~-t,

c

-x

~

+ (

~+1)

4

=0,

.<Mp+

2XM,C,

+

X.

=

J

1'"' .::.

e

=

~z.

M

Atx.)=

cf

ex

x.2.

d4. .

~)

"3l'X) =:

A'X.+

C-x.

I

e"'

rJ,j.f'X.2..

12.C.b)

~(

X-lt:~(';jlX),:-1~)+ X.-*cYl(~l?l),X.)-'J(X)=O,

'Otx))j

~

~(x)

=

c

1

~

r

Cz.(-e

~+

&c1>x).K.)

~

Jo~ .i..~

/tC>

O'W'l..~· ~

(11.b)~ ~[.i_ ~

~

~

'rf\I!~•

~U.1

d4

~~

~

~-44>"

,._

--4··- ...

1

8.(n.,

X):

Seo

_ea±

Section 3. 7

23

Section 3.7

A=

1 ,

B

=

-4,

c

=

12 , J)

= -

z

2 1

E

=

2

z

Section 3.8

25

(n)

't;i-~"-~~·-~ =-4~. ~~::

Ax

3

_+Bi

1

~ ~

'd ::

_A<i.)x

3

_+B<x)i"

1

•

~~·

x.~A'+x-'5'=0

1

A'=

x-

3 _,

_Atx)

=

_-Xl2.-+

_C

_P

3x,4

A' -

B'=4x.}

B'=

-?c:,

Bt.x)=

-X-2/2..

+

D)

M:S

~lx.)

= (-

i'-;2-

+

C)

x.

3 + (-

~112..

+

D

)£1

=

Cx.3

+Di' -

~

'·

Qt~' f~\fatnct5 ~d..

_~

j"W,rJJ

_{ol, \V(~)}

'*5

~

_{- -(\ ..}

~ ~ ~J

~M

~~

.Aj,Cx)

~M~~'d

1

Cx.)i.oo..~~.

s~

.fdL

ik

i!.r..

~.

7.

s~ it~~.

Ra:dwt

~

~

fk

~~)Lr

JA4'

·~

~

~~~~

t,

.N.·~'I ~4<n),~a°Xt4'1'1~P~·

x. ...

1"-~'-s"J =wf.

":lh

=

Ax'1r+l3-i

1

Ao~ ~

=

A<x)1'.3

+

!3t><)~

-1.

n...,;:

'a'

=

~'x

3

+

sx-

1

₊

_{3i ...}

_A-

_x:z.B

₌

H3~

...

A-~-z.13

r

~~j~~=o,~:r=,,'°d·

-j"=

3X'A'+

')(A-x~

B'+2.x

3

B

~d ~

iW

~ ~

O:Dt:.

~

(3~

4

_{A'+~-B'+~-(,x+~-~-(~-t-~)=Llx.}

AO ><3

A'+x'rs'=" 1

A'=4x.-

3,

A<'Xl=:2'X--~+c

3'X"'

A' -

B'

=

rox. }

B'

=

2

x. ,

Bex)

=

~~

+

'D )

M:J

'd(:t):: (-

~z.

+

C)

x.

3

+ (

x.2.+ "]) ){:

=

-2x.

4-C-x.

3+

x.+

J)\{1

=

Cx.

3+

Di.

1

-x.,

ll.a.

~.

Section 3.8

' · (c.1.)

m~u+ c~'

+kx.::

F.,

~Qt. ~

>

U

Yv\ =I,

.ife

=

3 2., C

=

CCJt

=

~

=

8 ,

.Q. =I> Fa= 10.

'r~ ~=

fk7M

=

,(32 CW\tl (l') ~cl (l') ~

'")L_

~ ~

x<t)=

e'Ft(AtBt)+

•o

~(t-+~1L)

~(32-1)'2·+ 82. ~ :-4t ~~'~

:: e

CA+B:t)-t-0.3J2.34:)(t+2.ss,).

~ ~ 0. o. x

RcitA.ui

~ ~ ~lo)~ ~1_{(o) ~ {O,n') -o.}

~

fo'1,

A,B)

;t

~ ~

~

:10

k

f4

~P<tJ

;-/

I

6 ₈ 10 I

X

~=

~ A=1,B=o.S"J~·~

-X.(o)::O.''''

~~ 'X.'lo)::-3.S?~.

TopeJ:J.u.oe.

~

\')\~ ~~

~tt ~~-tk ~

~

~:

> with(plots):

> implicitplot({x=(l+0.5*t)*exp(-4*t)+0.3123*cos(t+2.889),x=0.3123*c os(t+2.889) },t=0 .. 10,x=-2 .. 2,numpoints=2000);

Section 3.8

26

9.

Cll.)

~tt)::

'Xh

<t)

+

~~(Qt+~)

.

We..

CAN\

-k

'X,t/t) ::

o

~ ~~

"""

x.,....<t-)

~

~

~~

x"(o)=o,

x~/o)

=.o

j

~

~

~

A=B=o

~ o'>-~

~

~

tf't\. X"1_lt)

~

~

~ 'Xi*.)~

(20), 41.l

~:

X{O)

=~)+Ee<.:>~

=

E~~

u •••• ••

~~

... ••• ••

X-'fo)=~'

o)-SlE~~

=

-.52.E~~-

N"o~I

_\

~

=·

....,., + · " ~ _ + _.._ _ I + ~ · -"-'1 ,₁ •• IX (0) :

Cb) l~ Ab) ~-;:;...-~;;-~ ~ • :

~~-IA.QI ~(a!~~.t)/~

.

.

t

i,k

~

~

'X(O), 'X.'{O)

~

•·•··· ..•.•

:Lx.s~~

~

AA.Qr

""*-

~d

~

~.

=

"'pl*-~

10.

rnx''=-~('X.-~)

;Go

MX"+-kx=

--k~

'1f.:>J2.;t

=Eco(..Q.t+~)

ti. (cl.)

~

=-lc.=1

AO CJ=

~ii~

=

1,

Fo

=

2s-,

cCJ?.=

2.1ml:

=

2.

~

C=

oJo.s:,

1, 2/f-,

s.

> with(plots):

> implicitplot({y=25/sqrt((l-xA2)A2+0*xA2),y=25/sqrt((l-xA2)A2+0.25* xA2),y=25/sqrt((l-xA2)A2+l*xA2),y=25/sqrt((l-xA2)A2+4*xA2),y=25/sq rt((l-xA2)A2+16*xA2),y=25/sqrt((l-xA2)A2+64*xA2) },x=0 .. 4,y=0 .. 60,n umpoints=4000);

> with(plots): > implicitplot(x=S-exp(-t)*(S*cos(3*t) +(5/3)*sin(3*t)),t=0 .. 15,x=O •• 10,numpoints=6000);

A('---Q

I

V

t

'I

~-s.t..n.

~

I

=s

2 f 0 2 4 6 8 10 12 14 t

Section 3.9

Section 3.9

27

> implicit~~oti~=(l/2)-(5/9)*exp(-t) +exp(-t)*((l/18)*cos(3*t)-(1/6)* sin(3*t)),t=0 .. 15, x=0 .. 10,numpoints=9000);

.. r

Q •.

I

o. I ~I I o. I I 0. i i I

t

S~-sta.£.

d: O.S 0 2 4 6 8 10 12 14 t

We call your attention especially to Example 8, on the free vibration of a two-mass system. We1 return to that problem in Section 11.3 and study it there in tenns of the matrix

eigen-value problem. It is an important problem, and you may wish to give it added emphasis by discussing it in class, both for Section 3.9 and Section 11.3, and even comparing the two lines of approach to the solution .

.3.

~,

)(

1>Xz.>X3 >0:

m,x;~: -~x.;--k(x;X

₃

)-k(x.-x1-)

m

₁

~:.

=

-kcxrx.2. )-

~(~-K,3)

rn

₃

x;'

=

-kc~.-x.3)

+

k£~~x,3)

Section 3.9

30

To-~

~

s~5 ~d

A,B,

C,

E

F*

d~. ~

.:...to

.L...V.vi,

1

'1-'4..

~

OD~~,~!4f~:

'.Dx+(D-•)-J=5

~

(-Ae

:t+2B~~+

(-Ce?=+2Ee

2

t)-(-5

+

Cet+Ee7t)

==

s-di,

(-A-2.C)e"";t+(2.B+2E-Ef~=O

Ml

A=-2C-~

E=--2-B.

'f~,

?Ut)=

f-2ee-r+Be

2

'd

ct)=

-s-

+

c e

1

-2B

e

2

t

ce)

J)X,+

j1

==

~t

1

E:~~

·

(D'Z:~)$t

=

'J)(~t)-4-

=

C/:)t-4

~x,+JJM

=

4

~

('D'L-C))M=

-'}~t+'Dro)

=

-~~:t

. ..,. I Dd

+ . -

A

3t 0

-3t

ti

~ ~ ?Git)=

e +oe

-U;cc:>:t

+t

.

~ct)= C~:d:+Ee

3

t +-(-~

.

.

'f'o-

~

~

~

~~

A,

§,c,£,

~

~

~;Jo

~

1

tk

~cl, ODE~)

!'d

ii.e.,

f~

:

Jh:.+

j

_=

~x ~

.

(3Ae

3

t-sBe3t

+fo-~t)+(Ce

3

t+Ee

3

:t+~:t) =~

~ 3A+C=O ~ct

-38+£

=o

o12)

C=-3A

~ t:.

=38,.

~.)

xct)

==

Ae

3

t+Be

3

_t

_-fo-c.ot+±}

~(t)

=

-3A

e

3t

+

3Be-

3

_t

+-(c;-~:t

(:f) -x.tt)

= -

~

t-z.-

-'~

-4A

e

3t

+

2Be-

3

_r

~ft)=

1-:t- ;12!

l

xz.

+

Ae3t

+

Be3.t

(-?,,)

xct)

=

Ae'}t

+

48i\

it)

=

-fle"t

+

B<i;t"

CL)

x.<t)=

~

-*t-*Ae

7

+48€x, 'dct> =

tN-f,+;t:

+Ae

7

t

+Be-;t

(i)

x.ct)=

A~..fft

+

Bco{3:t

+2C+2E:t,

~(t)= 2A~"3t+2Bc.o43t

+

C+ Et

(M) 1G(t)=

tl3-t

4

-ft2.-f%;-

+

A~€t

+8cp.(3.t

+2C.+2Et)

"jCt)

=

-fgt~ + ~c,

tlf-

+

#t

+

2A ~€t

+

zBeo€:t

+ C +-Et

~

)

3t

~ .. (i)

(2'D+-3

x,+(21)+1)~=4€ -7 J)x.

+

(D-2)d

=

2..

d4Ji::::

2~ ~(x,(t))

t

1

t)+

3j Xlt)+

2*~(~Ct),t) +~Ct)= 4~ 'llf.p{3~t)-7:

d~2.:= ~(Xli.),t)

+

~(~(t),t)-2 ~ ~(:t)

=

2.:

~(£tl.ei1,ctei21, fX.Ct.)J~ft)J)j

~

x(t)

=

-2

+

txe=tt

-ls

e

3t

+

Ae

3t

+ (-6+2

C.)~t

+

(-2B-e)C(:>:t '4(t)= _(). -1-_{. 5}

~t

e

3

t

+-9.. _25'

e

3t

+

(~-3A)

_s

e

3

t

+

B~:t

+

C

Cef.:>t

_.

'1.

x,Ct) == G~(t+<V)

+

H~("3:t+'\P) X.:zJt)

=

G

~ct+ c\J> -

H

~(

..f3

t

+

'\11

>

(ll) X,(o)

=

I

=

G ~<P

+

H~'\lJ © X,z.(O)

=

I

=

G ~<P-

H

~'\/} ® ~:co)=

o

=

G

cpcp

+

43

H

Cj:)'\j.J

®

x{Co)

=

o

=

G

ct:><P -

43

H

t;p'\(J

®

(b)

~~~~~~~dt.,

~/V'A.~·

Section 3.9

31

X,(o)= I= G~<P+H~'P

~(o)::O= G~cp-H~"I'

m~~ G~4>=

1

_/2

x: toi ::

o :::

G

<-04'

+

43 Hee."'

a.o.

...i-.

~--+

H

~

"V::

112.

X~(O): 0

=

G

eo:>cp-

,(3

H

ec>'I'

c;

Cc>4>::

0

4'3

H

C(.)

'l'

=

o

~

c:P=

"o/=

1f/2.)

G=H=

1

_/2.

Ao x,Ct)

=

f

~l.t+ Tif2.)

+f

~(,[3

.:t

+1T/2)

=

~ <:.r.:>-t

+

tCf,:),[3.t

X₂lt):

t

~(:t+TT/2.)-

i

~ {"3:t+1f/2.):

tC/:):t-t

Cr.>"3:t

-0.5

P.eot~~~tk

~~~

~(~):

~({'X,:0.5~

Ctr.lC:l:) +

o.

5

*

C\1J(~Jtt(3)~ t.))

x.

=

O.Sf Ct:>lt)-o.st:~(Tt(3)~t)}

t=

0 .. 20,

x,

=

-2 .. 2 , ~=,CXXJ)j

CHAPTER4

Section 4.2

(f) x,'2:.3x,+ I ('X.+2.)2.. - -

R=S"

/ ~

-,'\

\ --iK---+---+--.-. I 3 ~/ \ ' '-/

Section 4.2

34

-7, 5" /

x

\ /

---- R=2

' ..._.

___

.,,,. /

Section 4.2

37

. 'd(X-)

=

A:

Bex-

xz.+

_ix,

3

-tx."

+

,~

x

5)

+ Ocx.~)1

~

~~

OWl.

..n.-.J.:t,.;,.,

E)t~

'7Cb)

(~

A

..o

ao

11-d

B..:a o..,12.).

( :f)

Fcfl-:

'X!'~"-

_"j =O)

d1.t.

~~&.. ~

(

'X."2.~

cl.4tf"(1('X-),

'X-,~ )-~(X)

::O,

:at')(),~

=~)j .~ ~

~

~~

oJrnt

~=O. fa~~

~p~

~

x

0

=2,

µ

~ -r=~-2- ~~ODE~ (~+2.)i:Ycr.)-°YC'?:)=O. N~J4

C6'W\'VW\~~

~( (~+2)"'2 -1 cLU,(Y<c),"l,~)-Ycr.)

=o, Yee),

~~~)j

~

Yee)=

Yeo)+

J){Y){o)r

+~"!to)r.2-+

(-fcr.

Yto)+

g

!XY)(o))

"i:3

+ (

i1sq.

Yto) -

~

8

J:XY)(o))

%: ,,_ -t

(-clo

Yeo)+

1

~

20

DO')(o~ cs+()(~')

O'l.J

i,...W~; ~C-x.)=

C,

[1+

~('X.-tf-iq.

(x-tf

+

~

('X.-1.)4

-fiz,

c-x.-1f]

Section 4.3

38

13.

"a"+

'J

=

0.

d

::<

l\o

+ 41 'X. + /J.2. 'X..,_ + £\3

x_3

+

a.<I X.ll

+

4t; XS" +-..

r

( 2llt-+ C,£l3X

+

l2.Ct4 'X-2..+ 20 Q5X.3

+···)

+

(q

₀

+

q, 'X.

+

42. 'X.-z..+ 4s'X.3

+

4q. %'f

+

Q5'X.,S"+···)

=

0

~o; 2~2.

+

Q.₀

=

0 __,,

a.i

== - a.o/2.

~I:

'43

+

t\_{1 ::}

0

-» Q3: - ll.,/G,

X2 : 12Clq.

+

Q₂

=

0 ~ Aq.

=

-£t2./1'2.

=

O..o/2.4

'X.,3: _{2oa 5}

+

4

₃

=o

4

a.

₅

=

-a.

_3/20

=

a.

₁ /120;

~.

-rk

_r__ N\fx) =Ct +lt

x -

Ai>x-z.. - 4 •

x.3

+

Aax.

4

-tAL

'X,c; - .•.

lfA>l) () 0 I 2. T 2't 12.0

=

llo (1-J..

'X,,,_+

$X.

4 -···)

+

~.

(X.-t;

~3

+

clo

x_'ii_ ... )

IS.

r~ E~~s,

_LI

x

=

l+~+~'Z.+···+~n-1+ .:£::...,~Sn= ·-~n. ~}

- l-'X. J-~

Sn -z. N l

Sa+···+ SN _ (l-X-)+ (l-'X.'2.)

+ ...

-+ Cl-~911) _ N-(~+'X.: ,_ ···-+ 'X. ) : _L _ ~ \-+ 'X+· ..

+

X\.J-N - N(l-~) - N(l-'X-) \-% N I-~

:-1-

- 1 l-~N _c.n jQ X.:Fl. ~- l'Xl<i 'fk~-

1

1- -0 l-0 = J _ .

1-X. N (I-x,)i. -i'"''" -

J

-x

1-X 1-'X

cap-1~1>.1 ~ik

~N~oO

a..o

~~

oO

D-Mtl.~ ~

~

~

~

~

~·

Section 4.3

39

NOTE

CAREFU~L)': ~

~

*

~

~

~.~~

1-0

Ex~t.~2./7f,

fen

.

.t?.~.

HAl\(W\' ,

~ ~

~

~~

)M~ ~ ~

2.1\tl

~A

3.J\<l

~

~

~. ~'14 ~~fa~

(l.e., Z:n)

kvv~ ~J

wt~~~

~M.~

~

to

.k

~

r.-.-t,

~ ~

-tk

-1 a.-tA -2

~

~

t;

o.

W.e.

~

ti<>~

~

~

>1=-1

~

~ Z_

1

tl{Vl+l)Yl~~+l'Z.~ ~~~

~

{Jk

V\+1-fo..c1dt;

~~,ik

n=-2~

..-.=-•

~

~ r::_4(t\+2.)(r'l+l)Q.n+2.'l-~

Mt.

?fM

~

~

j

tk

n+2 o.,..d. n+ 1

~.

.0...

+'-t

~

hi<.4+,

~,

a..4.tl.~

.w< cam

~~

~

3fo

ooaAM.

L

3

(~-? ~

ti._

₃

=

~z

=o._

1

=o),

kt~~

thL

-21<,

o

~

z:_

~~~AN\~~ Mt~ ~e.NJ~ n=-2AMtt t\=-1.

~,Jo~~-~~~

~t-, ~~

'2_~

"-<l

~"-14

L.;'

~

r.:

~ ~

~=t!.,,_=q3=~2=~.=o.

tk,

~ ~

~ n+~

L_

2 [<o+'l+2.Xn+st.)Qn+2.

+4a.n-3]

t

=O)

/.l-0

ln-t-

sz+

2. )ft'\+.n)a.n+2.

+

4ll

n-3

=

O

-~

n =-2,-1,

o,

I, ...

n : -2 :

Jt. (51-2.)

ll

=

0 ~ JZ..(Jl-2.)

=

0 , Jl

=

D, 2.

'f1~'f.3.I ~~

~

n...,t

(Yl.=O)

.w&.l

~

~~

dl

~'~

dk~

(12=2.)

~

~

""-

~-,J4 ~

~

~

"';J#•c.).J;t~-tt-..1

'1.=0

~

i'<

~

A.t..~4.;)

w

.e.:t

M.o.

~

..:t

~ ~ ~

A<t

fdl

~:

1J.i:.h

'1.=0

14

~

~

~

r-WJt'Nt

~

~

n=-2..

(n~2.)na.n+z.

+4t.ln_

₃

=

o

{n=-t>o, 1,2 , ... )

n=

-2.: ~

ao=

o.n.er.

n=s:

35(.{.,

+

4-l(,a,

=o

Ao Q7

=

-fs-a.2.

n=

-1:

-a, :::o

~

a1 =o

n=b:

48a.

_9+4a.

₃

=o

AO

a.

₈=o

n-= o:

o=o

M>

ei,=aner.

n=7:

b3d,+4£4=0

M)

q,=-tsa.q.=O

n=

1: 3a.₃=o ~ ~3=o

n=s: goa.

₁₀+4a₅

=o

~ a

10

=-1z,t:<s=~~

n= 2..:

Btt₄

=

o

J>-O llq.

=o

n=":

~'au+

4ll,

=o

Ao t{₁₁

=

o

n

=

3: _1sa.5

+44

₀

=

o

40 l<s

=

-

1

~

q₀ n= 10: 120

a.

₁₂ t 4-t.l.₇

=o

MJ

°''z.

=

-dc,a

7

=

~S"

t:t2

n=

4: 24 l(.,

+o

=o

/.)<) ' { '

=

o

&...

'ttt)

=

Q₀

+

Q,:tz.-ia

0

ts-

-fs:

£lz.t

7

_+;/s~t

10

+;

5Q.2.t 1 2. + ·· ·

=

~o

(1-

~t

5

+

_7'5

t'

0- ···)

+

a.2.(ta-~f

7

+1:..

t'2.- ... )

.$

5 35

gs

n=o:

a

1

=

o

n

-1L• /1 - _ l Cl.

-o

- ..,... • '-'5 - 5z. .3 -n-5· n~-..L"' - I a_

-

.

~

- "%.

144 - 2~'Z."'Z. 0 it:,

Section 4.3

40

Section 4.3

41

··''

.~o

o0 n-1 o0 n+l rfi, [~~

1

-1\1"(1

1

]k1.t

+Li

n-z.Cn~

-2:

₁ Cn?l

=

-2~; <n, (

c,-t4C2.X.

+

~c

3

~}·+1,c

4

x,

3

+·

..

)-(c

1x,i.-\-

c2.x,

3

+··.)

=

-2. (

t~

+

-k-X.

3

+ · ··)

x

0 : C1 =O X: L#Cz. =-I AO C2. =-1/4 'X,i.: ~c

3

-C₁ =O MJ C3:0

X-3: lbCq. - C,_

=

-1/S AO Cq.: - 3/L2'i1 >

AN-.d

A<J OV\.. •

'l'~}

1\12.lx):: M l'X)~~ -.Lx,2. - ~ ~4-....

A

('Ll.lu

di 4 12.8 ...i.J ~1 . (') A

:t .

N~TE:

-- -;-

nJ,

'X.~

1

'+1·f-

%d

::o

AO T"t. ....

-,~ ·~

~·"~c

i

~

0)

~

~

~

'd(X)

=

C,IOCX.);.

c2.

Koc~)) ~

Io,

Ko

a.n.t.

~~

~

~I;.

o'lAut

0.

4...J..

i

t4

~

~

~

~>

~J

44'

~

k_ .

~

~ s~

4.~. C7.wi

'd•l")

Ao IolX)

~

OW\

'JlX>

.AO

a__~~~

~

I

0

~

K

0 •

NOTE: Before continuing, note that the different cases defined in Theorem 4.3.1 depend on whether the roots of the indicial equation are equal, differ by a number not an integer, or differ by an integer. Since we will not be solving all parts of this lengthy problem, let us at least give the indicial roots for all parts of the problem, to assist you in choosing which parts to assign.

(!) r

=

0,1/2 (Q) r

=

0,0 (c) r

=

0,0 (d) r

=

0,0 (S) r = -1,1 (f) r = -1,2

(g) r

=

-1,l (h) r

=

-1,1 (i) r

=

0,1

G)

r

=

0,2/3 (k) r

=

0,1 (1) r

=

(1±-{3)/2

(l!l) r

=

-1,2 (n) r

=

0,4/5 (o) r

=

0,1

{P)

r = 0,1/2 (q) r = -1/4,3/4 (r) r

=

-1/4,3/4

(YL'2.-3n .. )t{>\

=

(t\-2)49'-l.. (V\::t,2., •.. )

n=

I! -3Q.1

=

-4.0 ~ 't1 : 4o/3

Section 4.3

42

n=

2.: -2Q.i.

=

0 AO l.l2.: 0 _ . .

n:3:

oa3=

a.,::.

4.o/3

~

4o=~ ~,Ao. A.~~-~J

Jl.=-1

~

~

W:t:>~~--S.t:t

st=2.

(~

~

~

*'

~

~,

~

1o'P~4.'3.I).~*~

(n.-z.+3n)~n

=

(n+l)«n-1 . (Y\:(,z, ... )

n=l:

4a,

=

240 M:1

a,=

a..o/2..

n=Ll-: ~go..4

=

5"£i3 Ad ~4

=

l{o/l"s>

n=2: 10~=

34.

₁ 00

a.2.=

3Q.0

/2.0

n=S":

40as:

c,a.4

~a;-::

a

0/U2.0

n

=

3: I g

a.3

=

4tl2. M a3

=

a

_{0 /} 30 it:,· m 1 _ . '..L~ ~ o0 t\+2. t.. J.. 3 3 4 .L ~5 ...L t;. , ..,

I

"f\.WJ. '~ ao=I ~ J

fd1

(X.)= ~o '4'nX.

=

X -1-2. ~ +20 X. + 30 +T,8 ~+ii'i:Ox_+ ...

To~~~~

"J2.(~)

MAt (41c): ~

₂

(x)=: K~

₁

lX)~X ~

-x,-•

L.a;

dn~".

~

~

~

.v:.:to-f4

ODE

~

x.

2

_[~-t

_2K~:1x

-K"c)1/X't.+

L~(n-l)(n-2.)GtnX.n-

3

]

rH

-x.~[~+

Kfil'x.+

:L~fl\-l)d."~n-

1

]-2.[~x+L:tl,,x.

]

=o

dt,

L;'Co-1){t'l-2.)~.,,xn-

-L:(n-1)cAnx,n-2.L:d.n'X.n-•

=

K('d

1

+~~.-~x~r)

(~i

1

+2.cl

₃

Xt.+

'd.q.

X3

+

l2.d.5X.4

+··-)

1

!K(

'X.t.~ f~

3

+:0- ~'f

+···

efl.)

-( -d.o

+

c:l2.

'X. 'Z.

+ 2.d.

3

x

3

+

3 d.4 /(. q.

+ . . . )

=

+

x3

+

-£

x.

4

+ ...

-2. (

~-·

+

ll,

+

tA.2.~+

d.3X,z.+ tlq.

x

3

+dsx.c.f.+···)

-4-x.'2.-3X.

3

-t~lf-

... )

X-0: <io-2.~

1

=

0

AO

t:l,

=

d..o/2-X1: -2tl,

=

0 M) ~2.

=

0

X.-z.:

~-d.2.-~3=

K(t-4)J

~

dz.=O,

AO

.wt.~

K=Oj

ol

₃

=anlr.

x.a:

t;,t1

₄

_-2d..3

-2.&.q.

=

O A-O

olq =

J..3/2-~t: 12tt

₅

-3J

₄

-2~

₅

=

o

AO ds

=

Sclq./10

=

3d..3/20 .t.k

~

dzcx.)

=

o

·:V'X'.l t.-'lC + '<'.-1(

Ao+~~ +ox.z.+d.3~3

+

4:1"'4+3,tJ

x.s

+ ... )

=

d.o

(~

+t)

+ tf3 ( x_2.+tX.3

+

i,x."'

+· .. )

2.0

H~~ ~

~

'dlX)}

~ ~

~t

~

'jz.00 ;le,

~

~

~

LI~.

~'it

~t

~

it.e

c.t3(X~+fx

+.1..'X..'f-f..···)

~

-A-

rnk

·

20

~

hfu.ro,

~··

I

,

~

d

0

=1,

~'

'd2f

X)

=

~

+

t .

o0 Y\T.

Cj.)

3X."j"

+:'j'+

"c1

=a.

'.()<X)=

~o

qnx_ 11.

~

[~ 3(fl+JtXn+Jt-I) ltn ~ n+Jt-l

+

2.:

(n-tn.)Q." iC-~n-t +

L:

C{Y\ X r\+Jl

=

0

~ 11

+

"

-+

2

00_{a xY\+st-}1_-0 ~ oO [ ] n+n.-1 a n-1 -dt, "'-₀ 3{n+n.)(f\+'1.-l)

a..n.

+

(f\+Jt)lln + ltn-• ~

=

0 ( £L₁:: 0) /:l-0

n= o:

[ 3(n+.n.)"2.. - 2.(n+Jt.)] lln.

+

4rt-l == O

-fO"l

n

=

0,1,2> ... (3Jl..-z.-2J2..)ltc,

=

0 ~ .Jl..= ~' 2./3

/M\c!

qo

=

an..fr.

E~

.n

~

~

~

('t\.t..

~.

Section 4.3

43

F~

.ti:'

.Yl=O: ~

*

~ (3fl-z.-2.n)a.n

=

-a.rH

fo't

n= 1,2.> ...

Yl=I:.

a

1 : -Q0

n=2.:

aa.2. =

-a,::

ao

Ao

a.2. =

ao/8

>1=3: 2.ll.l3

=

-Q2.

=-a.o/8

A.a lt.3= -~o/1'-8

n::

4: 40a.4

= -

a..

3

=

a.o

I

I" 8 M) Q4

=

a.o ; ,

'72. 0 ~

Ac,~a

₀

=1~,

'daC~)

=

I-~+ ~~i.

-i!a

x,3

+

'~2.0 ~4-

- •••

NN;lr,

.L.:t

st= 2./3: ~ ~ ~ (3n.-z.+2n)an

=-a..n-&

-ftn.

n= 1,2>···

n= t:

sa.

=

-a.o

M:J

a,=

-0..ol~

n

=

2.:

1'a.i.

=-a

1 M1 'l2

=

a.a

/so

f\

=

3 = 33

a3

= -

a.2.

~

a.3 = -

a.o/2.(,40

n=4: S,4_4:: -q₃ A<1 ~4

=

a

0 / 14'7~40 .J.c,

M)~ao=I~

o0 n

+

2J3 2/3( J_ _L i.. I 3 1 4 ) ~2.(X,)

=

₀1A.n'X

=

X. I- 5" + 80 X.. - a.e.,40 X:.

+

141g40 ~ -··· (tn.)

X.~"j"-(2.+3X.)~=0.

"J(X.)::

2:~t:tn~Yl+Jl.r

oO n+Jt+I L~ (n+n.)(n+Jt-1)4'n.~n+n _ L~ 2.Q.n.x,n+

-3L

0 £:{nX. =O en

r:

cn+nxn+.n-1

)t.tn ~ Y\+n - L.~ 24n

x

'f\+11. - L.~ 3 a.n_, ~ n+n.

=

o

~

r: { [

(1'-tS7.)(n+st-l)-2Ja.n - '3a.n-l} x,n+:Jt

=

0

(a_,=

0)

Ao [Cn+Jl.Xt'\+st-l)-2Ja.n-3a.n-i

=

O

fo't·

n=o,1,2,...

*

n~o: (Jt2_-~·z)a

0

=0 ~

.n.=-1,2.

~tA ~o=anlr..

Fu~~t}U

Jl=-1

{~;;;J-{ ~m~

dt~).~°*~

cnz.-3n)an

=

3lln-1

fdt

n

=1,2, ...

n=1:

-2a..

=

3llo M)

a,= -

3do'2-n=2.: -2ct2.

=

3 Q, P-0 tl2= -S0..,

/2-

=

'tto/4

.

.

n=3:

oa3=3a2

=

274o/4

~

t:{f)=o

~~a..~.~)

n=-1

~

~

~-

1\JJ.

~

~

~

~

~

n= 2

~

i4

otlut

~(I.fie).

n=2: tk*~

·

(n'2.+3n)a.n:: 3lln-i

-/di

n=1,2, ...

n.=1: 4a.₁

=

3a.

₀ A<J Q1

=

3a_o/4 n=2.: lOQ.2.: 3Q

1 A<! l'.{2.

=

3ll1

/to

=

"J0..

0}40

n=

3:

isq

₃

=

3ll, MY

a

₃

=

a2.I" ::

340

/ao

n

=

4:

2.8 ({

₄

=

3t<₃

'°°

a.q.

=

3Q._{3 /2} 9

=

~a.al 2.240

it:,.

.

.._,o0

n+2. '2. 3 3 ~ ~ 3 5 ~ '

~ > ~ ~o= I,~' . "ja(X.): Lo~nX. : 'X, -t-4 X +40 'X.

+

BQ?C

+fiiro"

+···

'fo

~

fk

~d

~

~'2.C~) ~

{41c..):

'diX)

=

K

"J/?l)

~'X.

+

r:

d.n'C, n-a.

Section 4.

3

45

10.

ca.>

~ Jl~>

=

x.:~·-+i.f.

2::

o..n.-x,"

~

o..

~

:t.

()"+ti:i'+

i-a=o,

~

o

=

'O''+r~+t:J

=

f

+r1·+~i ~~

;-10o.

~

M

c...tl

::Cl."~z:a.,,x.."

=

x,ct-.c-'f

L:o a_n'X.'t\. ::

X.

~Lo; ~x,""J ~

AN-t. Ac.t.

~(!-'fl==

'ln.

Cb)

p~

<ln.= Cn. +AJ.n >

iin.::~.,.

=

c.,.-AJ. .. ,

a...J.

lt~.'2.) ~

(IO.I),

J.:t...,:

'd('X.).=

A

'X-"<. [ ct::>C~k~) +A~ (@~'X.)

1

L (

Cl'\-tA.dn )~~

+Bx.

cc [ " 11 _]

L.

(cn-icll\) ~n

=

x,« [

A~o

!:

CnXl'\. ~

A.t

~)

!:

cl.n~Y\ n

-A~c

>

L

ol.r\-x. +A~~(

)

L

Cn'X.

+Sepe)

!.cn'X,n -

B.A,cp()

I:clnx,-n.

-B~c)

L.

d.nx,n

-B.t~o

!:

c.Y\'X.n}

=

'X,clf

(A+S) [ er.>c? ~Cn.X-n - ~c)

!:

dn'X."'}

~

+

~ll\-6)

[Col l

1.J.n?C."

+

A<Mol:

c"x."]]

((

.D·~

(c) X}·Nj.~-+ X ( \-r't,)rf + ~

=

0. Jt~-.n.+.n

+

t

=

0 ~ .n=

:ti.

AO ol=O AN\4' ~

=

~

rr~

.

.vdt

..k

~,~Lt~~~ C~[

1

~

1-i'k

~

f~;

fdt

~'

~

AAnlQ.

~

~c~~):c. ~'1 ~(~~K.,):s.

S.tt..k

~(X)

=

LooO (

C.Cn -

scln)

'X., 'I\. •

~

1

_f'X.)

₌

_L

_{(-sct\-c~~~V\C.Cn-}

_{Y\Sdn )x."'-l}

"ti"~)

=

L

[-c.c..i.+

Sdn-VISC .... -ncd., +(tl-1)(-sc .... -cdn +ncCn-n.sd.,)]

x."'-2.

?~~~-rkODE~

'X~,,~

L~

{

C

[Cn-z-~-j')cn

-(2n-\)d.nJ

+ S

[-C~n-~cn-(1'i.-/1')cin]

+~'~

.+

C..(ryin-~n)-S~+~)

+N\~

+

c(d~)-S~)} ~n 0 oO 'f" n+I +x~'~

+

2'

₀ [c(ncn-&.n)-S(ruln+Cn)]~ =O dt

) oOf

J

Y\ 0() Y\otl

ro

c.(nt.cn-2.nGl.n,)+

s

(-2.nCn,-n'Z.dn)

J?t

+

Lo[c(nc~cln)-S(nd.n+Cn )]X.

=

0 ~, ~ ~

L

₁

t

''

1X.n-+L1 {

c[{tH)Cn-ici"_₁

1-

S

fJn-l

~n-a"'° ct\_,

J}xn

=

O cit, .0

I.

f

c[

nt.cY\-2.n.d~

+(1'-l)Cn-tdn-11 +

s

[-2.nC~-n?.ci~

-(n-l)d.n_,-cr\-1]}

x..Y\

=

0

~

*

T..lu.

~

~~

AJtt.

~

ep(.D..,x) 't-n a-.!.

Ai,,.(J.,~)

-x."

~;

AO~ ~At:I:

•=O

~tlr=O:

n?..Cn-21\d.n

+

(1\-l)Cn_,-d.Y\-l

=

0

-2nc"-n-z..oln-

(n-l)~n-1Cn_

1

=0 .

,fort

":=1,2., ..• ,

~

Co~

c\

0

~

~

•

n=l ~

c,=

f-2.C

0

+a

0)/S ~a d.1

=

-(co+2.d.c,)/.S'"

n=2.~ Cz.= (2.C.0-c!o)/2.0 '1,.nd

d.z.=

(Co+2.tl.0)/20

n:3 ~ C3 =-(\'1C₀

-,d.o

)/780 ~d.3

= -

('Co+

l1d.o)/780

Section 4.4

46

~~~cnj"~~~

~~(10.3)~~~1-4~~

~

C,D,c;,,cl

0 •

Lit~A.tt

...

W..t.~ ~fk

Cvi''°'

~

aln'AJAOJ

~

C=1

~

.:D=o

J ~, cto.3) ~

"a(X.)

=

Ct=>(-k 'X.)

tc

₀ + ~(-2.C

0

+c:lo)~ + k,C2.C

0

-clo)~i.-₇1₈₀ (17C₀

-'9d.

0

)x

3+ ·· ·]

-~(~~)[

cl

0 - ~(C

0

+

2.d

0

)"-+ia

(C0

+

2.d.0)X.?..-ko ('C0+l7iAo )'X.3

+···]

=Co {

~(~'t)

[ l -

~"' +~~2.-J:z..

X-3 + ... ]

s

20 '780 } -~ckx)[-.L

x.+-1-'X,i.--'--

'X.1

+···]

5 20 7SO

+<4 {

C4:l(Jn~)[ ~X.-:$

x,i.+

~

"3_ ...

J

-~ckx)

[

1-

~"-+Jo

x..,.-

~~

x.

3

+···

J}

NOW""

M CAM.

~

-tf-.4'

ik

~

40,.tM

~'

1

t'-<.

~

(10.3),

~

c.,Jo

h

~

~1

C-.:t D.

Section 4.4

47

dt.

(n+i)~HC~)

=

£2n-r1)~fn.<~)-nfn_

1

c~)

Cn=r,2, ... )

at

nPn.c!(..) ::

C2n-1)~fn_

1

c~)-Cn-l)fn-2.Cx.)

cn=2,3, ... )

'7.

(O.,)

a/ax

~ -.L -2rz.

=

"'£

o0

f.

'('6) SL n 0 - . 2. (l-2.XJ?. ... Sl'2.)3k. 0 n. n

= 2:

o0 (l-2.i(.Y(.

+

Sl..'l.)t>~ ('X.).n. n (l-2.X.lt+ Jt'2..)3_{12. 0}

oo

n+l oo oo n+I eo n+2. -R:

L

₀ ~(~)Yi.-¥\;-

=

L

0 P~CX)lln - 2~

L.

0

P,:cx.)Sl

+

L

0 Fn'<~)Jt

L7

Pn-1("-) Sln

=

L~

Pn,'cix.)nn-

2~

!,o0

P~_

1

l"-)J'l..n

+

L.';

P~-a.

('X,)stn

.AQ)

-fori

n

=2)3, ...

'Pn-1

l~)

=

p~

('X. )-2. x,

p

;_,

('X.)

+

p

~-2.

<"-).

8.

(a.) ~ ~

4

,,..0.

1o

"1-p~

-t~(~:~)

=-k

2.(n-+!-3+~+ ~'1+··)

fort

\n.\<

~

"'oo ""2n 2. l 1t..

=

2. L ~ > AO ----

=

J

f

Pnlix,)l tJJ'i O ~n+I 2.0+1 -I I ,,_f:O •

1-10.

(Cl.)

Cl-'X.2.)d-''-2x'd

+ ?"()=o.

To

kt~~ ~=• ~;,a,~~~ 1l-1:t.

r~

~

ODE

~

tC2.+t)Y''+!i.O+tff

'=

o

-':J~

Yrt).

rp"'-

~

~

~ Jl~=o

Ad

.n=o,o.

1JJ!...

n=o,

~

'frt)

=

~a.ntn ~cl ~

(2.t +

t2.

)(2.42.

+

'43

t

+ ... )

+ (

2.-+ 2.t )(

°'

l

+

2~z. t

+

3ll3

tt.

+ ... )

=

0

;t.O: .2.tt I : 0 /.)() 'la

:o

t' :

iA2:+ 20.,

=

0 ~

a.z.

=O

:tz.: I 8tt₃

+'tlz

=

o

tK>

a

₃

=o

~

M) li'f\·

~)

Yrt): t.l₀+0+04-···=a.₀ MJ

Y.<t)=

i..

C9f

~

-tW.

~

tk

~

~

fort).

~d

:IO

(41b)

.W..

'1'~

4.3.IJ

At.c.Jc.

+k

,;t/w,.

~

Ito-

Y2.

(t)

=

Cl).0..J:

+

~7

Cntn •

P..d::t..j

~

~

t4 ODE

~

(2.t+;tz.)(-iz_

+ _2C2.+,c3

t

+···)

+

(2.+2tX

f

+

c

₁

+

.2Ci.t +3c3t'Z.+ ···)

=

0

£

1: -2-4-2. =O

t

0

: -I +2.C1

+

2: 0 AO

c,

=. -1/2

Section

4.6

52

6

0

~0. 'f~, TC&o)~ 2trf17i

>

~

o)

~ 60~0.

NOTE:

PJ\~d ~~

~

~

~' ~i{..i

t!/til:

1-

(1'R.1)

~~

~

ODE

-e+

t~e=o. ~

.

.

.

.

fdt-

~

~

(~.e.

1

J&l<<.1.~), ~e-e ~af-4 ~

~4

*AA.~~ (S~ss) ~~~

-e-+c~/i)e=o,.w4

~~

Sft)=,L)~(~.t+<P) ~~~~~'-~I~

~

~

7'

~

~ .,/~1.e

'T1=2rr

d1.,

T=

zrr~.e.1~·, ~

~.

l,.(b)

ro't

f<x.)=4-t.-OC'X."')

Md.~~~~ flx)-<l-/\J

Cx.2.

(aoX..:JO).

W..Jl)

fC't)-4

=

-q.~z.+ qx<I - ·· · rv - 4X'Z.. ~

x_,o . ../

I'

cf)

fort,

H(x.)

=

0

c~> o..o. ~...,.o ~ ~

to-

~

i-kt

Hr~> A.J

C

X- ~ "'~

oo.

tJ.J.l,

Hex):

:ix.'3-x.-+l " J 7'X}

=ix

~~"'o0.

v

x,l+4 x_"Z..

Section

4.6

2..

~)~

~

+

astt.~ ~

(2)-(10).

L.tt°MO.

~~(')ANA

{JO)

<J-0~ cf4a.)b),~. (')~

-r

c ) -

~

[

_L_

+

c-rf!

(~)'2.

a

(~

)4

]

u 1_12. ~ _{- 1 Y W3/2.) rCS"/2.) 2.}

+

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=~ftrn-t(tffe) ~ +2(-H~X-Hif) ~

4

-···)=J;(

2

_-tx~+tc;?l

4

_-···)

=

_{, rr}

_fi'X

(l-zC

_,

+~Q.-···)

₁₂₀

=

~

_rr~

:z...

(~-z!-+-~

_{3! s~}

-···)

=

_~~

r:!:"

MM"'.

S~~

{o'i

J_Yz.

C'X-).

3.

"foe):)= rr [

~<x.)

+

(}{-~2)

J;,<x.)]

~ ~ £~ [(lf-i+tn2)~"-i,+¥-~2.)Ct1~]

+

Os'-kz)~~ C/:)(x.-~)}

=#L~~~-~?l+Jrlr-11;&CA~-~+r

+

#¥2)~ (~c.r.>?G

+

t~?C.)}

=

~

(~~-er.>~)

:::

~:x. ~(?C.-%)

>

~

Wf. 2 1\tt

~

e<.>tA-8)=

epACllB

rrx

+~A~8~~fA-B)=~Acn8-~

4.

(<l)

Let~ ~

(4.1)

fo't.

iL.

CA4t.

~

'1:

..6

J.

'T'W

..0,

W<.

A.ul:tii

~1".ct:cr-iA

(XVJl,('X.))': X,VJV-1(~)

~

( ' ) J 11 00 (-1)"- l(.i.h2v x, JvlX,)

=

Lo

-k!

T1Cv+,k.+a) 22t+v y , 00

2.(-k-tvH-n*

~2k+.t1'-•

co

c-R.1-v)(-•r

~2.-k+2."-'