,.HitP 230 Z - Dlf!er&l1hal £"1 uc:rll~lj>1S

'Test#= 2 - Sprl//1 2tJCJ6'-'

Al1swer t/i// ~ tt-1c?S!/Or7S, An 1Arjus!/lied CJt11SI-t/er w;;1 -r-c?ce/1/e

h/f/e or no cf~cl/f. ;VA C~/c'vllV/ IdTS.' Or ,1c.1yy;1tAJasheeli. tare? a/kt-f/c?c/

F In-ida JAt'/ Un/v,

Ti !1F : 75- ?"Y7i 1-2 .

(/5) I Fmd fj,~ scJu h~t? o/' eac/' 0/ /he ~ II"",)I':} ODE,(4) .Y - r:; y" +- 9y = b w!11 y(o) = I J\!:/ (0)=5

(6) Y /1 .-;- Lf Y = 0 W I fir Y (0);- 3 43 ~) = - Ii ,

(;2tJ) :2, Find the.- 'qd<',o-Ic:t/ st.>/tA-I/dl1 ()( eczc~ c)1 the ,-/oM)tVil?C/ ODEs::" J

La ) Y II '- 3 Y i+ 2:J:= Lf X 2 (~) ~'j II ~ 2.';/- 3 y.= /2 eX.

(ie) 3, Hi'1d Yc aYlcl..tive tAe ,m/nt'ma/';;rm c7/ Me Yt? tltdt

one ,s:OIA'!~ -f--r;yfr edch 0/ _~e h/lowiJ OJ)£(t/) (p + 2}) -I- 5-).!1 =- :3X e sln(2x)

Ch) (])L-I-f) 2 (lJ ,- 2)3'y = 6' cos2X -t ;2 cos X

(15) 4(tJ)lJel;i1e whaf;1- m~al1s.ldr r;;-..;f;/ 10 6e ItneCl/'7 de;e/7de'YlI.

{J)GII/en that l{x)::=x /5:a SO/UliOI1vi (;/'+I)yil_t;XY/'f-LfY=-~

fo'uJ Ct s-ec'oi"7d hnearj,y Ind7H?ncli?nf .so/fAI,on.

(/S) 5"(tl}Wrife d"wn what !!vole 's- la/AJ says., r. I J" I j ' , I '-I )I. // 2

(b),ina a .IortlC't.elar SOlu (on C) f/"Ll 01.)£ Y -+y ;::: see.: .x.

(1£)6'.ft 6odYCJ/mass if1Y IS al/ac:Aej Ia a II(){)J:~-~/e /rlj

?VIlA Sjr/;1.J C(};,?sIClr7f k~. 2() Nm,-I c)n a htJ/"Jonfai -h-,c/t'onksS'$tcrlclce. The air rc?sist&'fj//c-e ~ tAt::: J7?tJi/oYl /g a/so)) V' IAlhete

,i):= /6Iv'sm-l. If /Ae s;rl'y IS ex/ended?, ell'? C/lrYlouni c-~/ 3'7Y/

~ll;me 1::::0 4n4 ihl?l1 /~f Ioosl') ,hndlhe /tJs/!/o/? 01

lAe ,bcltll ai all St16~e:lU?nf limes.

I (a)

MAP 2302 - ])//I~renl/a/ Zfiial;ns. J"

SaJufldrl 10 ?est #- 2

Ronda InlJ tll1/lI,

:5P,/J11j .2tJtJbI v

Y il - {,Y i + 9!i = 0 " , (D2 - ~;j) +- r) y -= ()

2. 2." ,

(D -3) Y = 0 ..~ CD -3) == ()'=9 D::: 3 (ft-v/(:e)

y = (CI + ("2 x:) e 3 x / s Ihe C!erlera/ so/ulo/1.I , 3X V

Y :::: [C2 + 3 {LI + (z x) ) e-y(D)= 1 ~ C, +- c'2' 0= I

!J f(c;) -:: S' 0) C 2 + 3 ( CI 1- (2' () ) =- 5'

c; = o. 5 D C 2. :::: 6- - 3 C = 23X

Y = (/f-2.x)e

(b)/1

C' 2. f \ .

Y +;-!I = 0 . .-j) +4-~If = () ~ }) =: :t 2 t

Y::: C( Cf)s(z.x) .f- (~s/n(2X) I!> /~e Je';1(-'rClI3d/U/fon. '. ':1::= - 2Li Sin (zx) ,+- ;;:{;;:' co s~,x)

y (P)== 3 ~ C1. I + C.2' 0 ::= 3

y "(0) := - 3 .~ -zCI ,0 .-1-.2 (2 ' / = .-'8' 3> G::: - ~

Y -= 3 Co::; (:2X) ,- if 5' in (2 x)

~ c==- 3

2(a). y il.- 3 Y /+ zy -= c:> ~ (p 2.- 3}J -/-z) y -==- ()

. . .'. CD - /)(lJ ~2)Y= 0 .~ Yc::::: C; e)( I- ('2 e/X

Try Yf' -= If X2. +- 13x -I- C. Thi'YJ Y; = zllx -(- 13d If ~ II. /. Z I

a r1. Yf' :::: 211. S CJ Yp - 3Yf? T- 2jj, -;. if- X 1;J(:?t"On1("'.J

'2-A. - 3(2 AX-I-B) + 2{A)(2.+Bx+C') = 11-><2

2 A =- L/- (cae II 1 x2) -?>;t =2

.- 6 A + 2 B -= D (('de?!/' '1/' X ) => B:::= 0

-;u4 -3B +2{"= 6 ((oel/o./ Xc:» ~ C'= -7

.:If' =: 2 X2-+- ~ x+-7. I/-(!/f.~/~? y ~!Jc f Ypy 2)<' C

C; e f- C2. e + 2 X 7-6 X -f 7-

,.

-

2(b Y1/ .

2 u / 3'

Y. ( ">, 2 . 1\ 3 )

i-J O.

-+ J'- = 0 ~.v +-2 IJ- -/:::::"

. > (.2J .-/) (fJ +3) Y =0 ~ !:Ie: == C1 e K f- l2- e -3X

h-y Yr::: /lxeX (be:.c,(2uJ'(' / is a -r(f?}fo/?J1u/li)-/ -1,/ ( (Il II)

Y d ' /1 (/1' i ') .)(/ I~ J p= )( I- t1 e a t7 ", Y,P ::= I(! X -;-2/1.;e .

u II. I, )if.s 0 j? + 2 Yr- 3':Jp = /2 e j;; e ('if me .r

[ (!lX --f 211) + 2 C AXI- A) .~ 3 (Il;< ) ] eX=- ! 2 eX// /1 e x .

2 x ,.~ f1 3 S . U '2 X 0 X--r /I ,=. / e =;;.- IT =, () J? .=-. v c

.Y ;;; Yc +.Y? = C~ eX+- C2 E.--3 X + 3 x eX:,

3 (to; 1(.0 '2+ 2 ])1-5 f Y = 0 '~ b == (- 2 + V4- -2D )/2 = -I + 2Z' (fW/~)

" }Ie = e- X'[ (C +(2 X) COS (ZX) 1- ((3, fC/X) 5i11(ZX)]

Th e "Wl/11/m a/;; / YY7 ? I Yp tI1alone -.S'I~(/ t-c/d 17 IS'

,'Yr = e -~ ,X~[ (Ae .1- AIx-) ('o-s(zx) .f (BtJ f 15; x) S;;1 (2Y)]

/;ec~av.r.e 3X IS ce /tJ(ynomJal 0/ Jp:Jree .1 cO'ld-lfzi

/5 a TJlJ I cJ / ?'11 u/!; r//[/Iy 2 ,

(j;}i I (lJ2-f- t)2{f) '-2) 3Y '== O-~ D = :l (31-Ime.J) ()r ~::: r i (fwlaz): 1 ,,' z.X ,

/. Yc == (CI +- (zXi-C3xZ)e + (Clf fL~-X)L6SX l-(CG- +C7x)J>/nX

NOIAJ o/;s~rve (hoLt b tos2X -=c 0, f [/+cos(zx)]=- :3I- :54:Js(zx)

SO (JJ2+1) 2.(lJ- 2) 3"y = C; CDSZ X -f- Z COs x= 3.1-:3 c:os. (;!..X)+- 2. cos)(

~/f/jt- e ::t tiS a 70 ()f v! ~ tLI!~//ct (.y 2. we -5>Jz 0 {/LId

fry (/10 Ct.?5 X f B() S //1 X), x2- 0: 5 a j?trf 0 f' !if'J- L /7 -/./ l' ,1 AlS' - // () cau::' r fZJ Y ( I'l e 2 c os X . 6 Sin c e.. 3 a n a

;3 ctJs(zx) are nol invulved /n Ye:. we slut-tit! ad).l

III vos(z x) -f- B; sin(Z)() + A2 -ID Clur YI' to cc~!-er('., -tJ. 1/ ;; ) fI I LL. II tI J/ , I1V' fhIL 3 COSl2)(' clna (tie 3 , II-euee /h(! mInIma

h"m t)/ 1Ii~ Yf' /hal C)n~'Shou/j fry /'J'Y,;:;. (Al>('0.1Xf- Bo s /n x).x 2. + Itl ctJsgy) -+ :B, sin G/<!) 1- A2 .

!I( a) Me fttnc/?o/?s i'() './ f~ are ilntPar!J de;endtJY11 if' we:

can hnl C:Ot1s!aNtJ.' C(j ~. 'J en Nh Ict P{Tc:' nol a/13r?l'o

5U[h tlzptf Ci !r (x) .f C2. ~(x) + . ., f Cn .1;,(7) == 0 ,

(b) II sccoj/Jd Itl7eClYf.y IHd,/,- scJ!u//()/'1 /sy/ven hj )12 =: if i(x)

J' .- f [Cltft)/t1o(;<)]dx

"

.'

J-f-fx h

'U = e d .- e x '1-1. .. .. X-. dx[lex)] 2 [X] 2.

J~ zln (x2..+t)

J' . /11 (Xl.-I-l)

..

2.

J ('

,

' L. )L

== e A X = e ~ tA.;x: -= . X +/ t:;{xxL X2. X2

-

J' X'f+2XL+ldxXL

iJlu! Ie

Yz (x) == V; lex) :=:

== J (X2+ Z 1- X-~?;{y

[.3 _IJX 1 f- 2X,-X =:

3-1:::: X -f-ZX'- X

"3

( i;Xf+2XZ-I.

5" (t?) HODke11'La tA/: The ;;n:..~ wlf;{ tAllllet- ~ /IZ)(ji~-7/e yr/p]

lulls back Is alrec t!.t l~rororDc;ndl It /he amount '(I

is &x/eHded!:om ;I.s.'nO-fu r~/ /~\f1l,

{.b) We Inot1,1 (.?ne. Yp = V;!/I -f Vz-!J2- t,jIze re ~ !?.!/2- are

fi/z/D /Ineclt'fy /i'lCIt'f' so/utO'f1I of the hOfll~, etud!;OI1

al14 Vj / = (;f<) ;;//!;;, :: I ~ v/ ~ I~/ ;0'))/ (~~ ~:/ I

JI-omrJ!f' Ft. IS' Y fl+ y .::= O. So lake !II -=- a::lJ' x ~ YL =. Sill X

( ./

I

(J

VI:= seclx:

SII1X

[/'!CCJSX SII'lX

)

:::

CIJJ X ~- SIf1 X t:05 X

- Sin x. se'c..lxClJ5l x f S'O'l2.x

R VI:::-

I

/1/50 vz:::(

co:;; X ().2) / 1 c.'oS x: SIt'lX

I

=--slf/lX .set' x l-SII1.X c-oSX

. 2= -L-. s-ecx = see x .secx

.=. - $111.)( !>€c)( = - fClrlX,s'ec,XUJsx

J- .f-ai"l.X' SI?CX -= - st?c X =. -tj Ct/S X .?

CDS x. s'et. X

CDS2X +Sfn2 x

/. VZ =-JSeCKi..x. =:Ltt ~ec x +fClttx)

, " Yf ~ c:oS X..:::: L -f 5111 X. tV1~:e~' x .ffltlty) =. -/ -f s/nx .111(see x+ feMX).C{)S.X'

6.'><=0

Fe (-Fz. ~.

- - - .. aX=3

..,mx .~ -h-G ==- -~v-I<x

,'.'11'1 Xi + ;1 X + k..x -= 0.. . ( , j .

, , , Lf x + /0 x + 20 X==-O .=;> x + if x + 5 x=-o

, ' , ]) 2+ L/--1) + 5" = 0 . =?:> )) -= (- I.f:t -./16--20)/2 =- - 2+ 2

So x{t).::.- ci2t- (II L'OS f- + 13 Sln f)

,', x{t);:: -2e-zt(/ltosf+ I3SIY1t) + izt (-AslI1t-l-Bcost)

:::: e-zff(B-zA)c<Jsf --(2# +/3) sin t J)( (0) =- .3 4 A- I 1- 8. 0 =- -3 =P

X&) =-- D =:r (8 - zA). i ' (2. A+8) ,0 ;= ().) -zt / ,)Xft= 3 e lc:osf + 2 Sil1t

,1==3

-913=b

EN A

lh,e 1~e51/()n vhd ~()f czsk ~y a ske IeIt at x(f) bul,If jf cluJ;1 f/z i~ j'.s !]()tv you t1JtJu.// do fha/.

3~ t xf!:)=- 3e-zt, /0 r e-:ost,.-L + Sin t. 2..]It L ~ ~!t - - zt, tI~ := 3J5 e lCOs, C{)S'ci( -f SIYli s~ino()

1\ =- 3/5 e2t UlS(t--o(J vJhl?,-e fan Of':=;.-;2. ~ Z .\ .

3 ~,

.0

~.~

, "I,

\'\"'

'- -:! 3-1'0e-ztIf'

tfs-)

:~\~--- -,, =,\~~-~ .- L - f, I -V- )..;=. ~ IT f- 7?lYl (2) 271+ TCln l2-

...

/ /' t{" -zt\. -3J5e/

II

jj,

(;I