3' '\/\I Q Q Q. /‘4 "-"<1 ---1' " -‘ ' - --~'- -1 .- ‘ -~ .-.. ::. u .. .' - -< r.>m r _+..,~,,,\_g,~.‘ 51:.-.,~,-...-4..,_..._,-,.,_ ,,_,,,..>._ ,,_ {V-5
I . :,.> . ' . " - :'
. Y~.;:.-.5 3:3 39- ‘T-... 4 . .7 ".'§¢,--__§_¢__:.)¢.pv,.-,1
w .‘, ‘,...\..‘-'-. -
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I
. ,_‘,-.-7 ._~
Non Linear Diff. Eq-" 50j_?0Ifder oneY
Am eq-L which is mat‘ linear, is colleql Norw-Linear} (See ch-‘10)'
Cor‘w5ic>le»-‘the non-'lineor diff» §eq- ffsb order‘ ’
1
I I
.
5
1
v
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'5
l
0
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1
./
02
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or f( I’ 015.91’ § = l Z ~ . I , A
‘ ‘ mi 5 ,
.' I »;. . ; . ..~->_-'._'.' I. .' ', - . x‘ - » - Y i
‘ A i- ~3=*;==4;~. -'~Y . 1' 1
' .; ‘I352-;;;»L~_’ .;;1.S"=[=: , V 1 ¢ ‘ v I‘ . ‘ V ‘ . n I > %
D ;Y-'1‘ 5v",',A|.'-1<'.~-1'-6" - ' ' I D - ‘_> :Jf.:14_-._ - Q . |/4 . .1-:1-:-<,w .»=~:w;' '--.-. . V _ .. . Y -
.~ , F . iv .
' J '.'.-->}~>¢g\_-" ",~\-. _\ 1- > _ . EB E. - L §.f>1-.§;'.»‘ . ’-‘ _ ~ I ~
1 _ ... ...,._,_ . -
.= = »v;;.-.1--;;:_; ‘ " » .-V »_,.A 4-z. - .-- l
‘ I
’ g — W 7-.E><a\fc§_5@
l.?£2!§‘J£!'¥£.!%sI 9% M1@@w»~k?@ L
TF1gs,'we usually , ‘raprasents NQm__[_i,.r\e;Qr ‘cliff e.q- -of U13
first Or‘>dE.r by }(X'9'p) = O’ whe_.'_Q' P__=_Q_L1~__l__
oil .’
We shall discuss the fouhr Le_¢A}-‘ml-ques to 5O\V‘e_ the eq..J(1’@#§),=o
\
|
.\{ | H \ ‘3 '
§\_‘Q
1 é .
DG»)6»)6+)(1%
S0|vobl<=_ for P ‘
Solvobla 3
Solvable ‘for ')(
ClOir"OuL7s QC]-' '
la” L '2, _3___ ___a_l_7_g_ 0|,’/0&0» éb L’
MW @>‘v+v "><»~,><w>=>m?)i-»(j"--Q<:9_.v"_‘<\j) £51.“ =xawU§i 5). "“
I)‘/ULCZ-bi L’ X,‘ - 3-» gi.::7 V2’ '\,., ' <' V
A “"3 H» J 3 3)-W 3 °%‘i*"‘:K ‘?if’"Y~ 2:‘
\’
"?1‘£i#;i?)L+(*“n"a”~n>%@‘-“=>‘v” ’ ~
§-11'
.'_1.i¢3,._i:.-
.;\f1
P
¢
B;
. 85 '
§@lwabe mt. P ~
' -, , ,. .1,-,_-§_g~j_q\.e7'_ .¢,_¢.,-,-*,- .~ -
.1.‘
3i|
}
"me diff. q. £1 3'9) = £5‘ 50; O be _,olvable. for P
n>
K‘
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O.
H-
if il; can be 'veoluc;eoL irwto 1-irwear factors-
: ' I ‘ I
r
1 - .
Examgpe 1,531+ W- Qty ; 0. '
501:” 2 2. 1
-.1
eI
..,I
3
1.
‘I
i1
Q
'i
XP-E] -0- 7(.?"EJ = O
=v (1P+,9)-(v¢P—a) + (><P—93 '= O
=w (1P_—3)[1_£P+3+\} =0' =v vLP-3 o or xP+g,+| _= o-
’ xP—~'j ' 7 1?+~_1+\.=Q4? =.. d ‘ d~J1 IT!‘ 1_ 2.. ,_'% at--i -:_'. H Q X -Ely-gk =
=> §i=<>_L£ => c>k9=_g_; -
'. . ,' H 8+‘ . 3 —~-»
F I I IW I ‘zv I’! .-.‘ "L H
-
F7‘ Tg = lr\‘L-rhjcA =_=7 >'"‘»(H+l§ =-|r\7‘.+ Inc,
1 I
=> lng = IQQ1. => |rW('d+l} ='lr'1 Qi‘ '
=v '5 = C1 =7 8+1 = ci‘=v H—=¢X=0' =7 x(5+1\,—-c=o
~//e"‘¢e the 9e.nero»I 501- is (9-c;>Q(xg +x—c;) =0 ‘
Exa 1 ’
I ! >LP-(x+X+&1)P+(x+x¢y_»g5){>_~;.3 _..-.-.0
» ' ;S¢Dl:~ ’ 3‘
v
fince. the _qLvenle.q- £5 5at£sf(eci_ by P=11 ' I - '
‘ v
\___
4'
X
I
I-/. 7H ' !'P‘_-.—‘|- (.3. u U P-—'>¢ = 0 ~ >'_F’—';J»= o
1; f+v_|o(y-dx{ .| vd _XdX Y_7 dg- = e1= .-__.=%__I 2! 1
|
V
' ='> P-I =9 50,,
~ ,. 1 In
. .1 = O X —X?-1*, -1I:‘ I Vv
X X El 13:7 v(P"-Q (‘XP_73-X1?-3'P +1131‘) : O ‘ A l
J /' ' B9
.5
otx A — .= vp 1¢.i‘»‘_
| |
1 . ':7 Jdfj = J0‘-X :7 I018 '=I'1dv7L' GU‘
2 J H “ST=v g=>¢+‘¢ =7_ El7=X/2+c 4
I
I
I
~ 1=»(P-O ('P—5O(XP—g;) = O
' P-1 = o ‘or xP_9 Q: O. _ k
~
' >
OUL . » dlx ‘j:
OUL I
C I ‘
' " ’ ' . =7 E:l~cx =oJ/erwcefh \ ‘ .
6'" 9e"<”-"°’ SOUS (‘J"X"C)(9-'1;2—C)(Ei-C1) =0
j % [email protected]@’ Em‘ Y %
TRQ diff. eq_
: " IQ V H=F(x)p) . '
=0 [5 £5O£Ol id be jg,‘/Obieor y '
' ' car-7 be ‘Pm: i ~the ' form l
. ' ' :7 |I’1\_-j '='|r1X lm '
=> \;;~7L—_c_ =0 :7 51-1/2A-c==o -‘F C
Exm 53 ' 5 % % V%9*PX =P1x“ .. Examle 5 =-P211» P —%l é)_y0h- H .P1X‘l_-PX’ @. SO|:-’ '
.
@ w,r.{_. 7L. we QQL QQ. @ w.7-.{;_ x_ we.
(ii; 3 2- ~ ci:T) S2E;i :2 T)l1_ :z:x:F) s2£I2_
¥ 7_ '
( 7
.02
' #1» P=F-\- 21P+1ci1'i__ J .
-7 P = /4xPz+’2xl'P %~l;_7>__x Q_L_P - ‘ ‘om' on
Z
;J‘:> QBOm I
2'" ’P<'*1>5P>—><0_1>2P\,;-9.” _ L T’ %§ = P“?f ! l otx * O -_ 211%; A
I ' *
.
:V (‘ZIP-r + p_p : O
\ _
4|».
é‘W/,.86!°'a/
%'°@
=> (P—1)[ XP (P~X)--9(P_1)] = O * +1?-a xg Q
;'
6’?‘Ti~1~V-,
TV. pl
.:
"*!‘¥“’7"'°’.“
§’"iiI* ' ' ‘ ‘ 1 1 c 1
»=> ('"iPX°)(1P+x9L—P§ = O A => Q: ___2P*-*' 9 .
=2
12
.,_.-._'.j:r'_~:
|
. , -
;- *1? , r
. I ' : ‘ I, I . . ..
O“ <11’ ;P(\—..P) ~ “Q; 1! . 3 ' “ r E
3‘ ':E" =57 —- - ~ ' ‘ ' -_;_. l,.2P7. -.0 or '2P+X9$LE —- 0 --> dx 2X \ ;
1 ,v X a-5: .__.-.--
Corwszldr, V _ ‘P "“"’7 i
=719LE=__
" — 522$ .£_ = '.‘.__ .__2P+x $2 = O "7 dP +(P—\\X PL?-\) ®
AIL is linear in xv, 3f(P)=-2P ’ '
u out .U(P)dP 2|n(P_-|) 1i SZL-E:-—_2C’L¥‘ ,',I'F'=-‘e, zz =e_"=(P—l)
1
n
I :3 = ‘*2 '!'T>‘l(-f ‘fie :> G-1.07-oil 2(-P_Q)LOlP =-(_£'PL\) dp
P X. /vlulkiplgimg @ by i-(:5 _[-F/ w<=_ gél;
:7 .iE-.-..—.. it f AZdl~* ; JP 25 ;< (P--v H5 +’2(P-OX_=_'_Eé'-
».1
..‘_-._._..1_..._C‘.
_>\~
I . _ _.
- * '~@+'~= ‘ MP-~»11= (—\+‘/P7d.P~ = lncxlz “ :7; o([x(P-02]» = My?-QdP ~
=’ P =1‘/>1‘ -§--@ Y1
' . 4 _=7 7L(P—V\) = .|"\P—P+C¢ 6/inwimiotimg frorvu ’@, @ - - '
=7 K_ C-P-rIr\P -
We 9“; 9 = ct C/XA 0?-01 @
|
1 2~ 7‘-tJ"CX-\-C,-=.Q_ 3-;P(-IT“
I H“
- I .
=7 X3 = Q11 __c Putgng value O{ ' 5'“ Q9-»@, we. get
_> C—P+|nP .
. (P-Q7" 8+‘) .®. @:' @ 8 I've, H19. 50]. OF ®
1» T \ 1 -1 ; wivaalné-FMX% W
.
TF.e. diff-~ . " ‘
i‘*1-CI jU€;y,P) =_0 £5 said Lo _be 5O|\/Qble {_Q,- X
fit c°""“°t b‘%fQ<1torizeok ’o1rwoL com be. Pu-E rm thg form" X = B1 P§
EXHIHPEQ X7, 1+ Pa
50!:-— |__(_ I '"' P + *i—--———-—--i~__________ @
-Dif/€»"erw££o££m . -' 9 QC] @ W1‘/‘-f. LJ’ W6?
‘I
» Q0] 1 * - —-~~-—»~ ~' / 91
-‘I
'.1
U
1.
r 1 ‘ 1
='rk"'_Lz!9L-E"\‘g—?- Vjl. .dy Pdu cu)‘ ' ~
,
I‘!~|
°.
II .
.1
=? jdu = j(P-\/P)dP
|
‘:7 [H = ~\-g ____®
7f1us ‘@, @- give the gemerol so\ 0} -Lhe gt»/eh_/ -:- ' D ' Q ' £73 O I £-1 '1 " \"- “M ' q p form" ‘mm
iiamls Egg, L '!'!!1~1; [Y-Am eq' '°} the tjpe U = ‘XP-rJ£(P) ’ » where P-.= %§_
‘ ‘K
is Called clo£rout’5 Equoh-on '. I .
Tlncorcm ,
emer-ol Solulzzorw OI lhe e_q. 3 =1p.{_S_(P) £5 Hy cx_‘_§_(q
Pram»?
I ?
~ :1=xP1-§(P3 -§@13i_{f&r€l:iOC£r‘W9 Q) w.,-.t If We get '
iH_I_ P" "~ P+1§K+j.(P1-ii}:-L *
' dx>
1
\
1 i <-L W \
! /
, , "’ >3“;I !+ (X + .f(P) _"—‘
:
...-n-.
l
f
\. .4‘
. 1,» 913 Remark '
"*"=*'%"-;»\
.
I . {he above theorern, i{ we c.or'\sioler x+ §(P3 =F\
we get 5 = -P§'(P)-1-.§(P)l '
if we. consider 1+}/(P) = I0 » ‘
D,-1 =__ _Jt’(p) pukL-{mg in eq-® ¢{ the above theorem
' TF5 pjor-arnehric £;q5- : ' I
1
S 9 I '
1 ‘ ‘
ac = —§(P)
t I \
= §<P)'"P§(P)' ' . ..
TQPfe’5e(§b‘thQ,'~§if\9LllOF SOL OJr~ -
7Fn's 501- involves no Orbiffad ¢°""5“°""' ¢°"e°'" 5""9'~“o' sown“
\ Example %
0-QETQWPIB Y A W Y
Find the general sol. amok
' . 5ir'\9ulOr so\-.0§' H =>iP+~‘qP“————‘-
E-Q_..@
:
SO|:- ." 50:“ HP-z'\‘P7(3"{\d =0 ' ‘
Find‘ the f]€f\£rO| 5QI- Ofd
Sirgulor sol- of _)_!%(‘_4J"P7Q = UP?‘-*@
T1: M IL? 'i§*"c»(ai1-out; eq; IL is Bt solvable for P, ‘d-I
. Gvé’-POI 5OI=- We com convert G). ite
CE.“
n_
ii4
1+
D:
' Iirgulor" SOI=-
.-.-
‘ RHOW U'\Ot7
[.
1 ' ' ' V. 2' l
I
, . » 2 d ~'“I {he cloirauf5'Eq- -2—:J—£?:i-i— = -3-!
~'rn i5
\
" Cloiroufs eq- Q5
Let U '= X 1 V: E/2
.-. _du =2xdx a 'dv= Qgddj‘
Now - - i V
u
.~>,' iii: XOW,__ . oLx- §jc:U-1
\1
\
x
‘ . l ‘ I ‘ )
r 1
x'0 _°®
’%,l.Z1%6
ea;4,,.57:
-_~;.,_»:.i‘n-:.:L»;,...-.u;_u;_
|.
1
-~
i»-
1
1 .,,
i r: Ll = ~H<_v1 ’ —
‘ ' v X
z»"
V
QV, /
‘ ' " ' 93‘, 5 ' _
T We can eh'mi'noL- P {row-@, <1: ' ' It-"15 C|Oir'0uk'5, eq-~ ~ . ' -'- ‘L . "Y,“ (_1)}'-5 1'5 gamer-Q\ sq 15.
._\
' ' '2. 4 \
I
"' ti"_= *3/l1("x)q/3 ¢ » V -= c“*.‘3 .
- - ~ - 2 2‘ vb '7 E! = C.)L + Q1 vrQq- gene:-0| sol.
1 7.‘ _:%liX' I I Y = - 1
' : 1. ‘ Eh"-QHIOP sol. ’_ -
/3:7 143 = ...3)(s- _. = av Zzf\(‘€ v u_Jn.,(%\/TD
»
\
=7 say‘ _= -31 | ~ - - . ,.
' 3 '-1' ' =7 V = uq +‘ 1 __ olv ‘=7 6H 9 + 3>L =_ o req- 5.-@9111; sol. ’ of I <=V— -an. I '
.‘. S,-r)9u|Clr $01 or above eq_ ‘-5
_'___ @=£¢<m-¢v,c'c<v) R
“ rvhare' / .
" f<°1)= W2 _-'- ,<c<=n =“2q/ .
. J/Qnce ® becomes, ‘ -
I " r ’ uh = "2q/ - >
, % , y = W1-1w€=-W1 1"-ff‘ @
1 L
I
II
.2,
‘E
I” I ’ ' 1 'I!‘ ‘ 1 * { I I
I WQ cL_or\ eli_mi'r\ote._<Iv_ .f~rQn-1 (3)
Y
» S;-nae ____ _U/2
‘ *7 ‘I ='--J/Ll,-=> -dz = _\/41%,
|
:7 Li“ 4' =.O feq. -Si‘,-jg. So|_O¥(:\
" v=~v%f
F‘
I
/ !
_/ 1
1 |v Q.‘
II iV
\~
r
F
\
\
r
\
I>
; .
>
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www.mathcity.org
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5 ..,
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5“; s
J.‘ 1' J;4i-\ "
_......~_.._. -------—--- -~——-~- - - 7
*4. V
‘ Malmiiyor ‘*3v
I ;\1 \J .
1
‘*3’ i\=...'|;_. ..>~_*_. . ',- -< .U’:J".
:'-:3!
1.-
/‘IYP-»
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.~:-.
KW...‘ 1..
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l"l‘?_9T""-T7‘-?-3.
~-E~.»“\‘~‘;"."
\
;\ ,»_
» ’ :15\\I.>‘;‘
‘Q12? E E
@w =
"61 \
fm,4
if!‘-I3,?“ :
1'-1* '~1!
.‘-.1
'f§1—_‘ " :
’ 0.
, -L->"
.2 ‘-'"LPI
v
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1- '.1-‘ .:¢ . 1.. _|
:7. ._ , ~
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Y‘ Y
2*.-
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1.,-
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iffg,-ii‘ 9
;§;-;~;r , V.
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..'~'.’-'_.¢"‘I"" 7‘, M1
9-.‘.:’,.j
1" " ‘
J.‘
>‘. .-= ~
ii
II
\I 3 . x
-n\__v‘;_..
;._‘; .
. _¢-I»
?_C§:I'I I1°-1-P-6 = O
5 ~ -
5 '35;; ;- , j0l;._
1
g . .' \ --Merging Ma" “"4 ma ,
L:-1 4 I ' ’
3 =v (P-2)<P+:n =0-3,
é’ ’ ' E ” ° °' 9+3Now
'—’7
=>j
% Q 3‘>L*-“J-C =355$} . . >8
1
I
A 4 . ~' -"5 (§l"2X*<l) (31+ y~Q =
'-’¢,’/ J |
' ‘ " ‘ "“ ' 2 1 I§§5¥.‘1. 75-_P+7(.f;}P....6_LJ =0
‘Ii Sol =
Z I 1 I I {
‘ I 1 I1
.-1".1:-.[;-L;
;¢’§‘
I zgg %°M '4-At '., ff. -- ‘i ,L mld ;;§,,_‘].;,.w____
,~ 2.V P-1P+3P—@ =0 ~
~ =7 P(P-2)+3(P—2) =0
= P-r'Z= =0r _P—-2 =0 ’ »
/1 d I OKL H= = __ = ~ .. v OM 2 _, <15
~ " 3+3 = Q
OW ézd-)L =7 <i'v = -3051,
GL9 =2]dx =1I<i9 = —3joL1= -rv ‘:1 2x c
=7 “J = —3x+c* E1 2x-— -C =0 ._
\//ence the genera, 50, » h
n._ '
' ' X'P—-11‘:1P+3x9P-691 = O
NJ
U:\J
‘K2,,-J
_’ 7 7(P<7¢P-131+ 3u (w(1>_ =9I J I I
, I - , .
krz-:11.
(*P+3sv =
7:) P = "33
' 4 < =v (xP—2LJ) =0 Dr
; =7 P 2 H/K
=> gag. = 19 /1%UL-I k ' OLE]
~ =7 'gQ= “IN/X.:7 _QL_g. = 1C_Q_( H
T-_'—.v‘|=> $1 =-QQ1
_§j|= 2!rwX+lm¢ V XZ: (“L L .iv hwy ,_:;;A v (Q! € _3§gg
— 171* WC, , I ii, ___‘l_1_*
O
“ | ,.$3)- '. : ~ . /
::) -= \q C-kl / =7 = *3lr“\X+ l\"‘.L, .
- xiv?" 9 = ax’ F" :7" ‘mi = l{3+I¢
{ ’- —— — nc'H_Ci __. 0 . __7' lrwg; _ l_ 13
. ~ I ‘=7 ‘ ‘I1 “' CZKI
\//emce the 3<-:r\e.r"o! 501.. J
@ 1531-+(’><i.-»u>M1>->¢ = O
[ I .5012’? ‘| I, ,.
' 'P1&l+vLP—SP—v£=o ‘
‘L =>-P(1_’§J+>'O"(T’H+>Ob=°
' ‘ => .(P"=1+vQ(P-M)’ =4 O
" =7 }_) = (IX:-5
=0
1'5 (_y—cx’)(_y-cli = 0
. =1. Pu+vc=o "or P—\"=o
vPH+.X=O ’ Pf‘ = O
=v P = J/5 ~7 d~‘:J --" '—-T = I
_ °__L§_ __ __x -- O-Mk "
=17 -= '--7Ld7L =7
:7 SHOLH = _S7“dX\ £7 E] )L+C,1 ,~~ ;_»7’.:~_9/»»\ _ -11 +¢
2 _ l . 8--)§"(; =
=7 »12-+§jZ *=C-V
- 7. 2 - I
; ; ;-:1; 7L.—r[\;!-—-(;1'[=0 ,. »-
Hence the" req-" sol- is _(1L1+§J5-Q\(§l—X—c;§ = o
A P3-"(>5+w+9’>'F+ >¢’g+xu’=» . 501:} ‘
firwce -Uni ‘give’-_n Qq. (5 X | la _,¢3:,@j_
» §@~L5r»*@@<1byP=>¢ 1 1* A >>m»i3¢,~2 - 2 /
o ~ -
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I LH_ X':!+ 15$
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O
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L2 O(P"°Q (P+iP—XEJ -5]) -= Q ' I 7L nu}-J J
I o:7 (P"_O[P_~'I) +..v((P*“J'3] T:
=7 (P117 Lowv uwv + x(1>-5)]: O
=Y (P-Iv mm u>+~_l+»m = Q ‘
i\~
v
A
_rn..vi
:24:-" '
5 ¥
E ."'5v
~»~znwr"
;;.;7;-=_:I::-:-_'...-~.'.;.'.v.;=.m§_{\__'.-__~L....__
'\'"U
M»-w7'4-
,5
L-_1-.._:,.._..-..¢
--.1
..'
KI:u
m_'@\\
I‘."1.1
P-"X, =:_ Q P.__Ej =_ Q P -1' 7L-1' fj =- Q u
iil m
../1!!
(F’—vQ(1P+*J—=@ - » ”“ 5*"
O I
95. Q-" '
OI‘ p_’_X-0-\j = O
- cU _-
. '95
_.._ = 3 — _>-+3 =-St (linear 1'05‘ .;.
:7 =—."!“)L dx ' :7 (ii .
=v§ol‘J = §1;<>L>x 9~ PF“ 3 = Q -
S O__L_';; SOL r4_u!£~.iF\§jir'19 thQ'o.bovQ e.q-bf/_1.y=’i :7‘ ‘:1 == ’f‘ CL‘ 3 ‘ u WC-L 9e_f, .
' 1
:7 7.\_-1-‘-)L1=:2C| K ‘ i ‘L'i|l"\L-:.)&_§_C_l vJ:7 29-xz—-Q =0
\
. 1-7 Mg) = lnQK+lnc;“ =7 adj + gédx =--—X<E2,LcLK
=7 9"ce:k= ° A=v ye‘ =-[1-é‘.-jédx- 1
‘—= -—A-3 -2-5.5+ L;
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. - F " :7 ‘I _ Ce Z O‘ E _- Q) .
1
J/ence Uqe rc-‘.q- 501- 1': (2y_1z-_c,§ (1-r y_|'_¢ Q) (5_¢_é) = C
. $ >c1>2+(g-1-ac’)1>_1(g_1) = O
‘ 5012- l
jinca -
=7 Imnj =_ [(43 (lei =7 O{(§]é(~) '= .*"iE7_LO(K
=7 LJ = pa‘ =7 Jd(Ué3_= —J>Léo£X
1 51] Pow
r
the“ SW9-m €¢l- is 5C1Usf|"€’Ok X X 9_l"XZ _XU+{by P = 1 7‘ 735’.-X
I O
¥
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P-_v(: Q ’ ‘KP~l~§J -—l'= 0-v_ O(\
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r[:21 2 :7 Jdjé =.51f;,Lx => -<>4_(v<\J7 = d>L
. . . Z
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P“ (X-2U'*3)P+3')L2Ia' v= Q
' ' §D|:-' " " '
\: I I ‘
I II
0 I
‘ HP+-.| = o
' =7 tJi§’+|=Ao
/= 97
~ 7LHP2'i’ (X-F 5)?‘-r 1 -.-_ Q
'Sv!Jl:§- -
46
vu-1P‘+vu?+:JP+ \' = .0 ~--.-‘v=> MP (EJPM) +(gP+I) = -o
|=‘.' (3P+1j (1LP+|§ = O - '~
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J, J X “T7 J‘-*1! = —-if-.>’_>£'
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I_, , ’ =7 -_-_,,, :7\,,_,,_ £1 lrwili5-I-21 -1: =-
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P2?-7L2‘:1P—3'P+ 3'23 -_= 0
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' =7 (P-vZa)(P'—:s7 ' =’ O
=7 ~11 —
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P—X2‘d=<>_' ‘ A%_V P 3 o
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I % , Qua »
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3-l4|%'1§'»§ll|3|l
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. all -= --7L/Q -I’
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. =7 31-I-7L1-4 C = O
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=7 295/ = obgH
=7 - "’“d = X -1 We
’ :7 Ir‘?! = le,-rlc J=7 ‘"8 == Icek
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2 7‘—E5(H+7Q—7(Qj-HQ
- = (1+9)(ZEJ-VQ
(J+1—) P +, (SH) (19 -U P_ -My (9-17 = C]
=v (L-f+X)lP1+ (an) (a+s~13P' + s(u~>O = C)
Y-* (H*>Q2Pz+‘ (H+vO5P + (91-vOQ9—vqP + my-17 = 0
'3 (‘d+17PU3*»°QP+3] + 9->0 '(;;+>QP-+:1]'¢=~@. ( [
.:74 (Qj+7QP+ 5] [(§j+ ‘QP + 51-1] =' O V
1
- > (E11-_'7QP-tr fj —_- or > Q1-r7L)P-,_.'§J_X O -
q91
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I _ =—_v (:|+>O ‘$2! =-(*9, X‘
&
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= ;_ ax-adx
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(M<>L>¢+ New- Let |\4~=19'; N=x+1 ILet M = 1+-‘d § N = "19.
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or 1>_7(_3Z-,=§jL'-V-C_ = 0
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l ~/7'e"‘¢€- @ becomes, 05-'/u?( req. 5.5o;,¢}® X __3P2
Ii U : P-J--3?} '-=. '-'2‘P3
I ' 1
we CO P‘ Q H771 (mo (Q ( '/(_'*1'7"1 (:1); C1
Since’ 1 = __3P’~
or =.".”'l}-P 3.
3' I ‘I I 14 I ' I
. =*2(1lZ)(11‘%')% A
= --2A(-_§')(1 /2;;
<-Ix
U->4
_Z
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