HAP 23ci2 -)J ,lkref1fial E1i/lahons F!tJTI'dt2 .In!! tin/VYest-#3 - hl/ 2tJ~8 T/H£: 75711in.- -

'.fwer aJiIP f-?je~l'b_ns- /In uyt,u;f/lieJ ~/1sweY WIt! yt?c&!..ve/I/lle ---

t)r?10 c~j,l. BEGIN EliCH Q()EST/~~!I/ A _¥P~!(l(G__PItGE.

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15' 1.110-l'1df/,e qel1era/ stJ/ulbn t:;/ /lze 011£ )(2)'1/ + !I = Qx2

lbv frans~rrY1 /ntj il Info a linear nOl1-hCn~()~~tJ~: ctJn~/'::ml -v v - ---

ClJe/hc/enf~JJG 1n !J aJ1~ t ~-

---

~Vl.o~j;ne wtaf if -yyzeU/1Shr XQ 10 le a sinquldr po;n! and-kr Xo 10v / -- -

e- a r7fIAICtY r;/n;;'AlaY 1(/111t t::J I ~ OlJe?-Y II + f} (x) !IIr- ~(J<)y =-().

(b tl hnL~~l'1era/ StJ I«i~~ pi flee OfJE Xyl/ ~x r+2..Y =- (J .

w~1i £{eat}[s) =: tjrs-a)._. /

fY?7nshYn7 -h +Jnd1/u;e ilze ?T~eri/~s o/;llf e

(f ) ct { t 2-CtJs t I(s ) .

.5'0Ive _lie. }; /IaIAJ/114 tJlJEs bif -!AS /n 9 fAe LczjJ/czce hl7 s;;rM; v v -2 f vi 1/ I

a) '!i (t) f- 2 f;/(j) = if e wt7A y(o) = 3 .y/I(t) - 2 YLt) -.3 yet) =- 0 NI# Y(o)~2 4 Y(o)::: -6,

(;;2,(;)

," For each t::./ iAe h ~u)j'nC/ 02) £" s: . h'nd Ih-e- 1Y)~c/a/-./ '

ua60n #tnd lliehrm at lINt) //nett:ty(y IYJd~ndeJ1iFyOb~n ;IA.~ sC'ries; sol/,,{ t/on ai)(:)d XC)-=- 0 .

al 2XLy/1+ X!:II-t-(X-I)Y =O_u (l) XZ !III -j- -(X-~)Y=O

I~) (,' Jlflnd the I;ys;f 4 ntJJ1-3ert? I-erms 0/ /it: power serres/

$DluilOYl t::Jf Ik tJlJE 'Y If + XZy =- 0 W 1£ /he/nil/a/ ~!!d«IOI7S Y(6) = 3 g. Y{oJ _?

-- - -- - - -

qi,

:j;1t?f 2302 - DllfeYC?YiI/a/ cYtAalonss()Iut-/;-;()JI7~ to lest#- 3

F?orld&! Z/Jil tI~;v.

Fall 20()f]--- -

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-- - -

11 1./ h L. /1 CI Z f) i t- /- - --~ I, il V}{e __a ~~~~+ _.!fILl = IX ~ rtA- )( = e anc:r

lJ =- d. Th~n X'-!:J II == lJ (JJ-l) y &- Xli ~ 2Jy .- -ilf--- --------

Sp- C>ll~ {)1J£-- be~~vnes_- ~(1J-/)!j +- 8/~ = 9.(c/)L

<~'- - i!!- Z-_lJ :L Vf-):/__J IE 2- t. flJ - '12.,)2!I = 7.e zi~ I u flz .L t/z . {/z , Vz

- - - --_Jll.- L - --~~ 1:;. e + Cz, /, e :::= C/ . X + C2. /n0) Xi .J- 2£- . 2..&. . d

,,~ - »- ~ A. e -'-- 7h~n !Ir'= 2/f-e ~ Yr:- lfAe

$°- our t)lJ_£ beU!Jnu?s ~A ez:C - zAe2i + fllezt-= gezt--L ~ - (:2'!Lt) If = 9. ~~ ,4 =- £1-- 5'6 Yp;= If e2:1:= 4 XL

/1/2- L /J). Z~ .::. 11- - J~ -f YP =- XI C I + Cz ..en X -t- "I X .

- --

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---

-

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D- JJ: a S i 'nq ula Y EO/'",; t!)/ !/ II ~ ;::(X)Y / -f- g&) !/ =-- 0V I" ,

II- _ai- ~clSt ()Yle ~/ the hv() h!tncl/ons !/(x) ~ ~&J is

?2t;t analyl/c at Xl!). Xo IS a. ret:1lAkz r s:/nCfJar po;nt-- - - - , v (/ /

if )(L is a st'r;JuLdr j/olnf an" bolA (X-Xo) F;(x) C?l1dx- Xc)'L~()() a ri!, a naly h-c ~!- X 0 .

I

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-_.- Z;)IIWehave X2-yll_Xyl+2Y==O. Put x-=ei:- an~--- - ---

~ =- it . TA-I?Y2 X!lI:=- 2J Y .a n d X2y //:::: lJ (:0 -;) Y.

~o -.!?ur onE ~eLt?rl1~ ))( JJ -/)Y - ;)!I + 2:/ =- ()

~ ~ ( jJ ~ - 21) i- L) :f ::= 0

~~. -_lJ =- (z 1- J f- - 8) / z =- ~:t 2 i) / z = 1-£ i

~ ~ Y(-t) = c; c:t. CO~ t --I- G et- S/n C

~- j/{x) -=- C/o X'. c..or{1h x) -I- cz-. x. So'in(In. x)

- -----

~~~- 10M:t j{&stj(s) =- ~[(eit+ e~~tiA3Js)-= yt:z Rtf!~tj{sif;t!ei{l(sil-- __,I - - ~~/- ._!~~/_~_-I- :I_!,;:: L (S/-i)t-(s-i) = s.

. :;l- l $"-.l. . Sri;- ,;;L5:2-~ iZ s;2-+1- - ---- - ---- --- -- - - - - - -----

-II . 2 2

[ J.J

JiJ,/6) 1.:X[f2a;s t](S) == &/), d ~[CiJst](s) -==-f rJ (5

)- (.! j I . . dS2. tIlJ L;;Ts S L-J-/

-- -~=--t-r°-;~:l-_- ~ ~:_~~t2tl~ is rS~ I - (S~~;)~J:::: 0: I ) , (25) - L/- S _._~? -s 2, _&2J(?~- -

- - - - - - -- - - - (5:2.+1 ) 2... Cs 2-+/ ) 2 f2z. +-I) 3- - -- -- - --- -- - -

--U- == - ~ S, (S ~I ) -I- 3 i S 3 -:= 2 ~ (" S L- 3) -~

II -- -- - (§Z-+() "3 - - ($ 2.+03 -- ($ 2.+ t) 3

--- --

- - - -- - -

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- -- -; - - - -- -- - - - 2-t - -- - - - -- - - - - --Y (t) -t- 2 y(j) ~ 4-e and Y{o) =3

!: /~.sl f!J/3 + ~ LifJ -=- 4- Yr~---z[3, ~~-_:_ot{YJ - (I{a) -f- 2_o!{!/J -- LfL~!2)

/ - (j 1-2) dt [;Ij;;= {l1~t-~2-- t. ~(o) = t/&-1-2) + 3

/ . X[Y] = Lf -+- 3- [SFzyz- --- Sf- 2: --- -

'" '\ -d - ztt . r -------

Li ",.

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(b)I

t ,

. ./

. L

/( ,

. f', . iI- " .

Pu t/rn

10-fhn

i

- - - -

- - --

- - - -

-- -- ---

!/I/(j;) - 2Ytt) -3Y(t) = 6_- (//l1d- c;{D)= _2 4 ytlJ~~-_6-

;t{yl/]- 2;!fy/? - 3ctfy) =-0 -- - -- - --

~L ~rlj J - S.//{D)- Y{tJ2~ 2 {setf!!} - 1/(0)] - ? ctpj-~- 0 - --

~2.-25 -3) .try; = s YLO)- -t:: Yt'°) -- 2 //(0) -- - -- ----

! + I)(5- 3) ;t {yJ-- --~ s ~ ~ -- 2 (z) - =- -- 2S -- 10 ----

eX[11] = 25 -2- ~ A. 4- g - -= A(sf-Uf- E(s-i ---0 -3 )(SfI) .5- 3 S -1-1 (3-5 ) (S-;-l)

- - -- - - - - '-- - - - -- - --

A~-I.i) -f- B{S-3) ~ 2~-/.O- -- - - - 7- - -- - -- - - - --

S = 3,/ ? /' v~s_- - A (31-1)= 2{3 )-~!? - =) - I!_~~-=!!: :: -(.I? / i/ e'S 13(-1-3) ~ -' 2t~)-[~ ~ - 6> - 3.. -- - -

£' 1115:=- 3/ ~+l) - '/ (:;-:3J - - - - --- -- -

YU__3er-t___~3t~_- - - - - ----------II

, i

~

L 1/ I /../. )(-> 2 I/, /

~@) Z >< y + X!I -f~. -I Y =O. ~ 0 2 X Y -I- X':f -: y -= 0

r w tit 6£ t/.e c<s.s:,;,"",,,,Ie d G uC/'y -euler eZ 1A4t/,;,,, ,1

- -,

fl1 i

5' (a) j I ,'. :2-r{r-I) + r - i = 0 will be /ke Indiaa/ ejl<fiftdn~__;l r 2-- r -J t2- _/~'- (:zr-+I){r- I) == 0" Sa-

r; -; / and Y2= - ,/z~~~,!ce 0-'2 ~ N; ~~~f Iwo

Itnearfx-lndej? SO/VL!ltJns0/ !A£ Arm- --- 1- '&>Q -;,--- - - -Yz- C>O ;y,

tJ/(!)~_x. n~ an. x' --- and_!l2(x)-:;:~_. ~- bn.X!:Jt!l.. t((J -/ _60.- -- - -- --- - -----

(6)11- ~l till + ({-if) Y = 0 '- - ~~ - X 2 !I'I - ~;jl::J 0

'NIl! b.e- -!he assodatC'~_0!!cIz'y- cukr £t,;~t;,,~.__s~-

1Ii.e-~d/cl.tJ{1 etVLP-£;;~ w;/I }e - r{r-I).:::- ~s-ff =- () -

~. i~!:- 15!.f = o. r - _~f II+/S)/~--- 5"/2 or -3/2

/. r; =-~z -_6\ IS-=- -3/2 . ~lnce 7;- yz- = If 6 !~~ we ;Jet

tlV'o_I//1-earI7 (h d:te'n!~;?! _~o/ul/o n.r 0/- ~e _.j;,r,!",':It~ ) = X5"/2.2: C?/1' ,x.nWtji C/o =- / 1 ar1 vi

Y2-(X) =- _A. X;~~f!c) _-I X-3~~~ bll Xl1- tA/fi bD~/'n=o

!l-Rre A is a cdnst~,.rt__tJhl!:lt- ~~ 01"'- ?71,ay n()i ~e O.

- - -------

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~. Y II + XZy =- tJ. {4I)J Y{o): "j- &. '?J:t2=:5~-kt Y= f 41'1Xn~/.:;;>E. n I oc n n =()

7hen y =- L. 71. t1n X - ~ 2: (n+/). Cfr1+/'X arZ~-- YI-O - - - - -~

II ~ 71-2 QO .) ny ==- L. '1'7(n-/). t:l1'1!-X --~ ~ (?1f-2) (rlf-I Cln+-2- X .11=0 1'1=0

s"" C*) 6~ cc.?;rle jDO ---

~ n ~ '> n~ (h!?){n +1)t:tnf2 X ~ - X < .c..., tit? X =-0

?1={) n=o

00 / '" ~, ,><; VI O<J n. 0-/-;--~t-LL-'-4-C/1112 + ~ r:[n-z.- X -- =h=-D 1'l~z

~_2 az' XO-t (,a;' x I -t-- ~~ {(pf-Z)(nn)c?nf2 +-- a,,-z.) >< n = 0

(~ zt/2 ==-tJ}bCls -= 0 4_f!!.-t~l0f-1) af2~2.. f- r::?n-z ~ ().I n~z'. 42- ==- 0, a~::o 0 & Clnl-? =: - a/?-2 / Cn+2)(n+l) n~2. ,)

;/(6)=33 ao=-3 4-_-Y~-5~ Cf,-5",

~ =- - C(illZ--fz)(2--tjL- :: ¥- 2-= ~ Vi ?'-_t1s =- -_0E~~ --ljLf.1/1

) - L 3 V s-f. J ~'< - 3 + S- X -r C),x + 1/.x - x - >( +- "~,,- -- - 7- y. --- ----

--

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---- - - - -- - --- - - - - --- - ----