Advanced Engineering Mathematics Solutions Manual

Section 1.2

Chapter 1 Section 1.2

3. (a.) s~ 4W=w-4x=y4s) ~~ 14! M=JA-tk>/&x =;v/l+tj'?... ~(Ila.) 1 tlNJ. cub), fC(.)& =A /:l-0 T~e:: ~6=~ =&=-t'~1+~ 1 i. u + .a

1

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

Ov-t\d. &/~ ~

Chapter 2 Section 2.2

'()

= c11+~,z, .

A

TCe .

A

1

Section 2.2

2

Section 2.2

3

I

Section 2.2 {~)~:

1

4.

1

Section 2.3

Section 2.3

2.. (b)

- sx R cl'C

,i.(;1:)

=e

0

L

t

(Jo

s; E«)l e E;:) d.t+Lo)

tJ t.L .~1): .. O t1:1_~( t

A.lt)::

e

ti..

. \ Ix, eR't/L to i:l'C Ho)=

E

.

Jwo

.

e-R:t/L. (o+J..•0 ) =J., 0 e-Rt/L.

-fct-:t,)

t'(l-e ~

;t

•

.Mt)=

, -Rt/L

) Hoe

S~AtSt: .let)+ ErR \

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

1

5

.. Section 2.3 ·

6

1

~ R~(/Jt-) = A~otCf.JW-~ct>cr.>tJt). ~~ () g:: toe-{,()~ ~ - w-k :: ;,.,. o.g.) 1. 0.00312.. ,60 00312 rn(t) toe- 0 ·003'72.! . 2.:: 10 ;t ~ .t 432.C, ~J 2 0 3 aJWl 0.1::: 10 e ·00 '1 t ~ ;t 12.37.9 MJ10f. -Jc.;t rr./ J -70~ " - - ,_ 0 ~~ '7. m.(t) = m0 e . o.Sr-0 'floe ~ .If{_= 0.003188. ll\M.\.) (). 5'JP{o o/., e-o.003Y-88T ~ rp ~ 21~.4 ~: 1 1 12•• (ll) Ynt\f =rrt~-Cl\rj (\j(o)=.0. ~ N" +~M=~ (~-d\Jut~~·)Ao Ntt) e - c.tt/111. ( efcctt/m ~ tlt" t A ) = e -Ct/M ( ~ ed:/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 )

=

=

= =

e ·

=

=

=

= .r

s

s

=

With. X.J 'd ~ 10 t, l\f, ~ f>Ct) - - 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 .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\vv\tl ~. ~ ~)D..

:t

-~ l'Y\~ ~M-dt ~~~:tr\~:~=: fi_2. v~ DMN\~ P~ ~~ L t ~ rn ~'f1-/'£

NOVJ,

2'.r\.a

'£

x

c

*F "' ~o x~

j:"')' /£

~/'it ~

1/'t'

£.

'Po.~~~ t' ~ "'°4 a. ~~ ~

1T) ~ ~

'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). 13. 'j)O'(o (H+P)'J= (N+Q)~ ~ Y~, ~ ~ ~ ~+~=}(+~

O't

0::0 ./

Section 3.2

14

CHAPTER3 Section 3.2 L (b) a. Att w LD ~it~~ LI) .40 ~ ~ k ~. NO. ~. (b) 1x~ x.z.+x.) x_z.4Jx.+1J x.-1}. c~-z.+~)- (x,z.):: x

=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[X 3 ) 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:

4

W[e '~ ... leo."x]=

eG..J.~

a.,_e"'x

~.e~'X.,··· a..,,,_er...,,:x.. s'dm'l:>7~~10.'t)ih.O

: n-1

a.1 rt·~ ea..2 ~

=

•.•

·..

~·~

:

a.,><

:

rH

Q.i,X

I

l

·· ·

a.,

a.2. ...

n-1

tln 'X

a.i e ··· a.rt e

e

ea."''X,

ea.."~

n-l

TH 1 • ··

l

O.n

•

rri.t. .4tt.n.

~~ ~

Q.

v~~~~c~ · . ·

1

n-1

E~~ 1'1 > ~ to.~) AJ ~ a.a't)tll"l(. ~~~~~)~ ~v (~&ea

.

.

a. 1 a..

a.n

~~~~~x) ~~· ~~~rr~s.2.2 tkt &-t a.;~ Ant: ~ ~ [ ~· ~ ... , e"· ic: 1 .i.o. LI.. s~ >{. ~

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

"'1~.11~ ('1'~3.2..2.), LI.. • • ~

= X-_,~ ~ ~ ~ 0 cn""'d ~. t\~ C'.11~3.2.2)J LI. s~ ~ ~ t"..o ~ ~Jh. ~) 'iJ[?Ci'X.

J=

I 2.X.

'2.

aJ'\l_

j~~;o ~rr~3.2.?f: ~~a.~~ ~)~~LI. C~) LI.1ra 'P~3.2.lt.

4.

Ch)

W[~'2X)co2x,J

=

l~2.X. I 2~2.~ -2~2x = Cp2.X,

-2:po MJ ('11~.3.2.3)

1 'ii.t.~;

LI.

Section 3.3

Section 3.5

Section 3.5

17

Section 3.5

191

Section 3.6

~

.

t.(b) ~= x, ~ A-1 =o

cc;

20

...\=I) /)=Ax,) 'dl2)= s-= 2-A ~ A=s/.z ~ ~o.

={A xK+ 13~ K

O'Y\.

o< ?G< o0

A~K+ 8(-~JK oY\-oOU:,

xD~= T>Y. ~ x.DCxD~)= J:fY ~ O't ~2 D-z-'d Jf-Y-D'/ = DlD-1) 'I.

=

=

xD ( x2.D-i."J) D D(D-1)Y 3 'X3 D -;j + 2~ = 'J>'l.(D-1)Y D(D-1)'/ -x, ])1~ = Jf(D-1)Y-2D(D-1)'/ :: DlD-1)(:D-2)Y)

~

3

~AO 6Y\,.

9.cI->) ~=A+B~JL 1 ~'CJl1)=0= Bin,~ B=o~ ~t.n)=-A:1iA.v\. p{JiJ=4'i.=A) ~ ~fn>=q52 1 10. Cb) J..l. =A+ B/Jt .U. lJ1,) =3 = -B/5l~ 9 B= -3J2.-:- /)0 ,U{Jt)= A- 3ll~/~. ~ 11.cb)

=A- 3.ll.f/Jl.z ~

A= 3Jlfl.R.z. /llO M(Jt):: g;p_f (f--:}) s.....Jc. o= At'Jl)x. -a'=A+A'x., ~"=A'+A'+A"'X- AO 1 2. 1 xC2A'+A''x)+ ~l~A'~)-/}h=o) ~2.A' +(2'X.+X.7..)A =o o1t, ~ A'=P', ~ ft 11 B A' B-~-2~-t, -x ~ + (~+1) 4 =0, .x).K.) ~

Jo~ .i..~ /tC> O'W'l..~· ~ (11.b)~

1

8.(n., X):

Seo _ea± 1

tn.

~

~[.i_ ~ ~ ~ 'rf\I!~•

~U.1 d4 ~~ ~ ~-44>" --4··- ...

'Otx))j

,._

Section 3. 7

Section 3.7

A= 1, B=-4, c =12 , J) =- z2 E=2 z 1

M

~ex)= c1 e~+ x.lf-4x 3 +12xi.-22.x.+ 22

23

Section 3.7

24

Section 3.8

25

(n) 't;i-~"-~~·-~ =-4~. ~~:: Ax3+Bi ~ ~ 'd :: A 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);

Section 3.9

27

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

Q

~

IA('----t 'I V

~-s.t..n.

I

2

=s

2

4

6

t

8

10

+exp(-t)*((l/18)*cos(3*t)-(1/6)* sin(3*t)),t=0 .. 15, x=0 .. 10,numpoints=9000);

.r

f

0

> implicit~~oti~=(l/2)-(5/9)*exp(-t)

12

14

t

S~-

sta.£. d: O.S

I

Q•.

o. I ~I o.

I I

0.

iI i

I 0

2

4

6

t

8

10

12

14

Section 3.9 We call your attention especially to Example 8, on the free vibration of a two-mass system. We return to that problem in Section 11.3 and study it there in tenns of the matrix eigenvalue 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. 1

.3. ~, )(1 >Xz.>X3 >0: m,x;~: -~x.;--k(x;X3 )-k(x.-x1-) m 1~:. -kcxrx.2. )- ~(~- K,3)

=

rn3x;' =-kc~.-x.3) + k£~~x,3)

Section 3.9

29

Section 3.9

To-~ ~ s~5 ~d A,B, C, E F* d~. ~ .:...to .L...V.vi, ~ OD~~,~!4f~: '.Dx+(D-•)-J=5 ~ (-Ae :t+2B~~+ (-Ce?=+2Ee2 t)-(-5 + Cet+Ee7t) == s(-A-2.C)e"";t+(2.B+2E-Ef~=O

di,

'f~,

ce)

f-2ee-r+Be 'd ct)= -s- + c e1 -2B e 2 t

j1 == ~t ~x,+JJM =4

.

..,. I

Dd

+

A=-2C-~

1'1-'4..

E=--2-B.

2

?Ut)=

J)X,+

Ml

30

1

E:~~

.-

~ ~ .

?Git)=

·

~

(D'Z:~)$t = 'J)(~t)-4('D'L-C))M=

Ae3t+oe 0 -3t ti -U;cc:>:t +t

= C/:)t-4

-'}~t+'Dro) = -~~:t

~ct)= C~:d:+Ee 3t +-(-~ . . 'f'o- ~ ~ ~ ~~ A, §,c,£, ~ ~ ~;Jo ~

1tk ~cl, ODE~) !'d ii.e., f~ : ~

Jh:.+ j

_=3 ~x

~

.

(3Ae t-sBe3t +fo-~t)+(Ce t+Ee :t+~:t) =~ 3A+C=O ~ct -38+£ =o o12) C=-3A ~ t:. =38,. ~.) 3

3

Ae 3 t+Be 3t -fo-c.ot+±} ~(t) =-3A e 3 t + 3Be-3 t +-(c;-~:t

xct) == (:f) -x.tt)

= - ~ t-z.- -'~ -4A e

~ft)=

3t

+ 2Be-3r

1-:t- ;12! l xz. + Ae3t + Be3.t

(-?,,) xct) = Ae'}t + 48i\

~

-*t-*Ae

it) =-fle"t+ B (ll) X,(o) I G ~x-z.. - T4 • x.3 + Aax. -tAL 'X,c; - .•. () 2. 2't 12.0 4 =llo (1-J.. 'X,,,_+ $X. -···) + ~. (X.-t; ~ 3 + x_'ii_ ... )

=

=

2

=

-a.

IS.

r~ E~~s,

_LI x -

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

Sa+···+ SN _ (l-X-)+ (l-'X.'2.) + ... -+ Cl-~ 911 ) _ N N(l-~) -

:-1- - 1 l-~N 1-X.

N (I- x,)i.

_c.n -i'"''"

-z.

N-(~+'X.: ,_ ···-+ 'X. N(l-'X-)

N

clo

·-~n. ~} J~

) : _L _ ~ \-+ 'X+· .. + X\.J\-% N I-~

jQ X.:Fl. ~- l'Xl.1 ~ik ~N~oO a..o ~~ oO D-Mtl.~ ~ ~ ~ ~ ~ ~·

Section 4.3

Section 4.3

39

*

NOTE CAREFU~L)': ~ ~ ~ ~ ~.~~ 1-0 Ex~t.~2./7f, fen . .t?.~. HAl\(W\' , ~ ~ ~ ~~ )M~ ~ ~ 2.1\tl ~A 3.J\ .AO a__~~~

i

~ I

0

~ K0 •

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 (s) r = 0,0

Section 4.3 (YL'2.-3n.. )t{>\

=

(V\::t,2., •.. )

(t\-2)49'-l..

n= I! -3Q. 1 =-4.0

=0

~

't 1 : 4o/3

n= 2.:

-2Q.i.

n:3:

oa3= a.,::. 4.o/3 ~ 4o=~ ~,Ao. A.~~-~J

AO l.l2.: 0

_

.

.

Jl.=-1

W:t:>~~--

S.t:t

(~ ~ ~

st=2.

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

n=l:

=(n+l)«n-1 .

4a, = 240

M:1

42

~~

*' ~ ~, ~ 1o'P~4.'3.I).~*~ (Y\:(,z, ... )

a,= a..o/2.. a.2.= 3Q.0 /2.0 a3 = a0 / 30

n=Ll-: ~go..4 = 5"£i3 Ad ~4

= l{o/l"s>

n=S": 40as: c,a.4 ~a;-:: a0 /U2.0 n=3: I ga.3 =4tl2. M it:,· m1 _ . '..L~ ~ o0 t\+2. t.. J.. 3 3 4 .L ~5 ...L t;. , .., "f\.WJ. '~ ao=I ~ J (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,-• dn~". n=2: 10~=

34. 1

00

I

=

fd1

~~ ~

.v:.:to-f4

L.a;

ODE ~

x. [~-t 2K~:1x -K"c)1/X't.+ L~(n-l)(n-2.)GtnX.n- ] rH 1 -x.~[~+ Kfil'x.+ :L~fl\-l)d."~n- ]-2.[~x+L:tl,,x. ] =o 2

dt, efl.)

3

= =

1

L;'Co-1){t'l-2.)~.,,xn- -L:(n-1)cAnx,n-2.L:d.n'X.n-• K('d 1 +~~.-~x~r) 1 3 3 (~i +2.cl3 Xt.+ 'd.q. X + l2.d.5X. 4 +··-) !K( 'X.t.~ f~ +:0- ~'f +··· -( -d.o + c:l2.'X.'Z.+ 2.d.3 x 3 + 3 d.4 /(.q. + . . . ) + x3 + -£ x.4 + ... 3 3 -2. ( ~-· + ll, + tA.2.~+ d.3X,z.+ tlq. x +dsx.c.f.+···) -4-x.'2.-3X. -t~lfX-0 :

'O''+r~+t:J

=f

o..

45

:t. ()"+ti:i'+ i-a=o, ~ o =

~

+r1·+~i ~~ ;-10o. ~ M c...tl ::Cl."~z:a.,,x.."

a_n'X.'t\. :: X. ~Lo; ~x,""J ~ AN-t. Ac.t. ~(!-'fl== 'ln. iin.::~.,. = c.,.-AJ... , a.....J. lt~.'2.) ~ (IO.I), J.:t......,: 'd('X.).= A'X-"C~k~) +A~ (@~'X.) 1L ( Cl'\-tA.dn )~~ 11 +Bx. cc [ " ] L. (cn-icll\) ~n

= x,ct-.c-'f L:o Cb) p~

= x,« [ A~o !: CnXl'\. ~ A.t ~) !: cl.n~Y\ -A~c >L ol.r\-x. +A~~(

)L

n

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) X}·Nj.~-+ X ( \-r't,)rf + ~

.

rr~ .vdt

~c) !: dn'X."'}

+ ~ll\-6) [Col l 1.J.n?C." + A~ ('X.).n.n

0

=L P~CX)lln - 2~ L.oo P,:cx.)Sln+I + L eo Fn' a3 =o ~ M) li'f\· ~) Yrt): t.l 0 +0+04-···=a.0 MJ ~ ~ fort). ~d :IO (41b) .W..

t' :

~

Ito-

Y.~

~ ~' ~i{..i t!/til:

-e+

t~e=o.

52

o) ~ 60~0. NOTE: PJ\~d ~~ ~

1-

(1'R.1)

~

~~ ~ ODE

.

.

.

.

fdt- ~ ~ (~.e. 1 J&l,c)

OQo =0 ~ 4o= t.=I: (q-'3)tt 1=0 ~ q1=0

-k =o:

'k.=2.:

=

./

l~ 'lJ=3,~, ~~ ~ Jt=-v=-3.

an.er. = -ao/8 a 3 =o

(1-~)~z. + {{ 0 ==0 ~ 1{ 4

-k.=3: co-,)a. 3 +a.:o M) 'k=4: (1-~)a.q + a.2. =O A2.~.

v

(~E"~5)

fb. ~ ~ ~ n(3.w.lt! f~ {di E~~: 2.rwl..ul., N.W.McL~~ trrp~ I

~~ ~

~Ao-I-ht

ka 1>~ B~ ~ t'732.> ~ ~ ~ EAJui_~ 11g1.

-f.Mt ~ 1

(0..)

0..

~ ,,,,~ ~ ~

41\L.........,.

~B~-f~J (b)~~"-t-4~~~ ~~~-"V l (tl.)1 .. Y=o. =o M>,..;..(SO)) Q=IJ b= £Jz./~, C.. =O, ol =2., v:O ~ 'd(~) = ~ 'l 0 (~ ~·12-) .oo '.de~> = A~(~~ ) + B'fo ( ~ ~ ) 1. ~ "fc-x,>= AJ0 (~~) +BYo t~.f.R:). B~ ..tx:t ~ B=o. Cb) Y~.u~·) =o . . Let r.~ k i4 ~.1 J;,Cx) j .C:~e·~ ~lx~) =o ftn. n =1,2, .... ~ ~ ~ ~J ~ (,j ant. ~ Jr;r 2.CJn~ = r. .. > &'?.J

[~c..e-x.)Y'J'+fC.i

{..jn,::

t,/'{ ~n

L~ ~-X.=~, ~ (~"a')'+°f%.'d 0

(n=t,:l, ... )

.~ i.,~2.405', ~ 2 ::::s:s-w, ~3 z

8.C,54)

.it:..

~~~~ ~~ ~~ Y,..ctt-) (J:to.k~ ~+k,

~f~Y),~ Y.ctX.)= A~3:t 1=L f '1 L ~ ~3t] =3 4-: = ~ . (C)

Lf J: :t Ct-1:.) 8 e-3'C t:l.'!:1= q ;t8 * e-3:t1 = L{t 81 Lf e3t 1 s""' O

-

-

t' f FCs) Gts)1 =[' {t2.

11. (C)

8\

s~

s

=+'

I

s+3 ·

~31 = Lfts1 = 2. ii :t"' = ,~±+ ~ f{t)~ (t) =:t'3 1

Section 5.4 1. Cb) I 3x,'+X.='1e~~ 3 [si(s)-1/o~J + X.Cs) = !

= ~ ( __!_ '1

5-2.

-

_J_) ))()

S+t/3

j..2. ~

tit)= k..(e2t_ 7

e-t/3).

i~: v~ +k ~ i'.J::t}:: 8e:t-4ert + ltio HCt-3) (e 2 H- e:t.-3 )

1-

'X, ~3~ +2 'X. : JOO£ (:t-3) , 1

1

XCO) =4, ~/(O)

t_, _

=0.

=

xc.t):: g et-q e t +too Hrt-3) ( e et. . AO ~ro> ~o+ '*-•12.) =0.02. (3000)(0.01)(2+0) =O.IS2 *3 = n f(X +t/2., ~ 0 +-f!2./2.) =0.02.(3COO){O.Ol ){2+0.075").. =0.13'13S -2.

,t2.::

~q. =~ f

0

(X 1 > ~o -t-l.3): 0.02..(3000 )(o. 01)(2+0.13,35"f2.

'j 1= 'j 0 + tC-kt+2:k2.+z1t3 +k. 4 )

~ l=

=2.14o~s)

~co.02.)

=0. 2."2.1~ =2.1'3,,7

I\ ffXq~'):: 0.02. (30G0)(0.02.)(Z.14-0l5) 2. :::: 0. 2.G,l ~~ ~2. =~ f (~+ ~l:z., '(j1-1--kt12.) = o.o2.{3a:DXo. o!)(z.14-0lS + o.13 lOOf2. = 0.3t1g~c, 2 *..3=1' f(~.+ h/2.,~ 1 +!212.) =o.02(3oooxo.o3)(2. t40l5 +o.17 3.313'8'7'744':3. ~r:,~J.:70 120D;Jgan} •• ., 3'® X(12D?>) = 3.a5"2107&lflf

t"'-

rJ

X.(l800)

=

X,(2.lflro)= ~(300())

~

3.~,72.38'7~'7

3.,~2.70) ' ' 5

= 3.~CJ8372087

'X(3b00)= 3.~~~'3'1~2.

~~ ~'?8 ~) ~ E't.~'ra.),~~)

tv(

At.t.~t4 ~

~;t~ IO~oJ.~: . 8. En""' Ctf Ao.~~~~ .1' ?~-?t~o)~:to c.. To~U ~ .1.t C=l ~ r=2.. ~) fo't ~ rn~... '°" -It M- ~ . -h= 0.1 4'= o.of ~:; 0.001 Cf,t =i:k-z. = 0.01 o.OCX>l o.oooooJ +-- ~ H~ c, o.5Cf!::: o.s~-i.= o.oos o.rxxns o.oooooos +- ~ 4 ~f, C-h'l.P== i.h = 0.0001 o.ooooooot o.ooooooooax>f-t-M~~ ~. W.e. ~ ~. ~pAl'Nl. J-[~n.+Ol-ltJ ~(1Cn)+~frx.n,~CXn)J-k] ~ "'· C...U. it" F ~n+~11 ~r\+.Q,) 1,::: ~~('Xn,~n,%:'1.), 12.:: -h~(Xn+L>'dn+~,, '2r\+~1) In~~~J fortn=o, = -h(4~ 0 ) C0.2)(1.fXo)::o ~. :: hC-Jjo) co.2)(-5) =-I i:,:: -h,(4)(~0+.0.,) (D.2.X4Xo-I) -o.B 4 =-/t(-('()0-t-.k, )) = (0.2.)(-(5+0)) -r 'd• =~ 0+ t(.k,+,k.2.) s+-£ (o-o. S) 4.'1 (~ ~!) ~.:: 'r:o+ ± {.el+.Q2.) = o+ f (-1-l) =-=l )(,:

*•

= =

=

= =

=

4·ih t5~ R-K: ~ ~ ~ ~ ~ -I!,= ~HXo,~o,lo) =(O.'l.)4~ :: 0 Q1 :: t~

=

fi,):

( " ) e (0.2)(-~ 0 ) =-I

-k:z. =-hf (~+~2.J ~o+-kv2., ~0+Ja12.) = (0.2)4(~~ .Q1/2.):: -0.4

) :: (0.2.)[-(~0+i1/:z.}]= -I 'k 3= ~ f ( ~0+~z, tj +-k2/2;e +.l.:z./2.) = (0.'2.)4(l: +Qz/2)= -o.q.

~z.::

-h 'C

"

0

0

0

l3 = t~ ( =(0.'2.')[-(~0+!2./2)]~ kq_ =-hf (Xa> ~o+-k3, i:I) +.!3) ={o.i)lf-(i:-0~~3) =-o:u,s .lq. =-h ~ ( '' ) =(0.2.)[-(~ 0 +.f:3)] =-0. X) =4 "i:.('X,,], ~ {?::(~), x,) = - ~(?G )) ~ (2): S:, ~=~J ~=~([3,5,IOJ))j

r

~ ~

c~, '(1(~,, %;(?(.)]

=

-.2..080'7.342.IO

3

s- 4. 800851531

[

4. Ca.)

~'Ct)= F(t,~ 1 ~ 1 i:)

= G( i!'{t} = H(

~ict)

Eu.k:

-Xn-+l

JO

='d- t

;

l

0.71~7583300

'X-(0)=-3

)

.,

o. '"~ 85381'4~

-4./882.,7,bl

= i! ; ) =t+~+ 3(~-~+I) j

••

-2..2.'732.'4-3S'77

%:(2.): 0] >

0 r,{O):: 2...

"j(O):::

=~n+F(tn,~n/~Y\,~V\)h

~n+1:: ~n+ Ge

t:f\+, = ~n + Hl

)-h

"

) -h ~I = k f C:tn> ?lYl>~n >~Vl) .Q• = ~ G( " )

2nd.~~ RK:

~He

m, ::

11

)

1'\+l

=

Z;n+1

= z., + tcm,+2.m'2. +2.m3+W\4)

'dn + 1;( .ka +2~z. + ~3+ .Q4)

)

82

f ~(X), i:lX.)]

Section 6.4 ~~MM,{.\. M/'~ ~

Sk1. ~--

Q.NI.

C5"~ RK ~' ~~ "(j,,~ 1

~ X2,~2.,r2. fdl ~ ~ .VX~· E..u.k: 'X.t = 'Xo+ f(.t0 ,'X,01 ';) 0 ,~ 0 )-h = X.0 + (Aj 0-I)~ -3+ (0-1)(0.3) 3.3 ~I =~o +Ge .. >""- = ~o + ~o~ o+ 2.(0.3) = 0.b '"i, =~o + H( " )t =~ 0 + [t0 +1'X1,~ 1 ,~ 1 )-h = ~ 1 + (~ 1 -l)h = -3.3-t(O."-l){0.3) -3.42. ~2.:: "jt -t (;-{ '' )-h.: "Jl + ~I -R :: 0." + (3. 8:(0. °3) = J.'74 ~i. = c 1+H< )"1. =~.+ ct 1 +~ 1 +3(r.- 1 -"j 1+1)}h =3.8 + ro.3-3.3 + 3(3.8-0.'1+ l )J(0.3) 8

=-

=

=

=

='·'

2M O'Nlvt RK:

= "'-)~lt) >ett)]),;

.tzet - 3-.tt ~ ~(t 1 ): X.(0.3)::: -3.\'185"12.'70'1 J 'X.(:t2.) = 'X.(O.b) ";), ~( Uc~>,'X,) =Vt~), t9{C:;=1

J.,t' = 2.'X.Lt.-

Yi=

L[ Y2.] =o:

Y, ;

(\j

N"'= 2.'X.r-r-Yz.

85

.~(1) ~

=

'f,'Co) =.UlO) 0

;

Y2 to)= o

;

Y.jJO)=fffO)= I

"Yr=.w-

L[Yr]=3~$(.: j Yr(o)=4t w'=a:xw-Yp+3~'6; Y/ro>=wto)=h. rr~, ~(O)=i = C,Y;co)+ C'l,'1.ltJ)+ YpfO)

rp~)M.t.v.Y~ tt=i,~~ c,=o~ J.J.d. cto~ _If. ~ 1-0 ~ Y. t'X.). d(t..)= 3 = ~. 1. (2.) + c'L y2.(2.)-t 'tr(2.) = c2ic2.)+Yr], ~= ~, ~ =~ ([o, o.5, 1.0,1.s, 2.o]))j

r ~

,

1.0

2..o

t.S

0 , 0$222.0~tf52., 1.2\,JC,ll«BO) 2.,Z37,23?2., '7.G,3~1~012.I

Y2t'X)=

~~

O.S

O ,

4X- ::

(Yio Y, i.w)

(rcq.{C Yr~1, 'X.) =

J.Ar(X,])

~ ( Mf(X), 'x,) = 2* 'X.._• >JS"(~)- Yrx) + .3• ~('X..))

=

Yro) =1~ wto)~o JJ f YtX-), .w1-6

v

.

Section 7.2 CHAPTER 7

Section 7.2

90

Section 7.2

0

I.----.---'----'----'--'-'--'-'----'-'--'--'-----'-'-'.......

0

A

B,C

B,C

91

Section 7 .2

ce) ~. (d) 'X}:~

]~:-JE

~': -2'X,} 6J

Ao

~

'd?.-+x"'= c. 6N-~)X-bY•o ?~ [ ~D-Gl){D-d.)-bc])'=o J>-a.1 -cX+ CD-d.)Y=o) .J- ~ [ " ] X=o. ~' Y''-c1.+J..)Y'+ ca.J..-k>c)'/= o ~ :: e>-t · ,\'l.-Ca.-rtt)A ;-c~-bc.)=o) )\= a.+ct ± ca.+a -i._ *~-be)

=[0v+d.i({a.-d.)'L-t4bc]/2.. v' To~ C3,Cq, ~ ~ 1 c Cz.) rt {Jo).,.,,Jo te;): p~ ~(,a.) 11

~ . >. 1C, e»•t+t.:i.Cz e>-~t :a.Ci e.\,t +a.ti e>-at . +DC3 e)\1t' +hC11-e>-2t ~ ~ eA,t ~ e>iit AN. LI) J:" ~ ~ A,C,:: a.C,+.bC3 ~ C3 >.,·i/·)C, Ai.Cr.= o:Cz.+hC11 C*=(~a.)Cz.. P~Clo).WO (.,b)......,,..U ~ ~~.

=(

~ €)~ ~~ S ~~~. 5. (b) lit ~«EC J ~ol: S. (5".I) ~ (lfo) ~

4. No.

rotiOJ

~C-~'s::: ~(l(°c-;s)+!(Xs+~c)

x's+~'c =-fcxc-~s)-ll(xs+~c).

E~J:.M,

.

u

3

X'= (flci.-t-~c:-J8si.)x_+ (4c~1?cs*11si.-_fgc~)~ ~'= (-.Ii SC-4s;-Hc --Ii sc)x +(--11o. V.&J.) .t =fg ~ 4-Y[t+11t.t.=4-2,m[Y.8 +11/8: I..; o 3/ci8/9)

f':

OMJ.

'tz.= 21fi~j~;l;1.-tll:: 4V

-Ji~ (l?. >o c OJNl. ?/ >0.

~,::t:

1

~.

.Ci:: Cz+Ac. +ACz.t ~A ~r-to/J) ~NA~. 'XCa+ Ci.t fJ

7, fJ'tlw\. ...e..1w- {19,;..)J >.1~0 ~ 1\2.-+ -ca ~ t:~co) Ao

"J "

~f>~~AA~: 8. ~, ~ co.~-'X. S~~~ ~~)...

fdi~.~JNf;~~W(

~~~~~ ~;.-r~ ci.t ~ ~ ~ ~~·~. 9. UAC.. ~ =( tti- cA ! 1-(4-..-tA.)...... £ +_lf_l>_C 1I2. . (a.) A= (2 +,m)/2 =3,-1-+ ~J,JP.J.... s~

'cl =l