!#"$&%(')$+*,*,.0/2143,56.0.+798;:2<>=@?ACBD<6EF?<6ECGH=JIK<6LNMDBDMOIQP
SR .,TU.2VWR>X&YU.,YUZ,X&Y[32YU/\Y^]&X&YU_\YU`,YDRSZ,X>a
bdcfegihkjmlon>prq6sJtusmvxwKyz&s4s@t{tu|}t{~@€U
 ‚„ƒ
…‡†‰ˆJŠ4‹ oŒQŒŽ
db w‘~u~{’U“”yxpuq6sm•@t{’Upr’U•Kw‘+–+|‘’—˜pw™piprqšso|‘tu’›‘’—x“O|}tipuq6sm~{yS~{pus@v
œ˜
œ˜ž Ÿ ¢¡¤£‘¥
œ˜¥
œ}ž Ÿ ¦§  ¦©¨ ¥
ª¬« B#­\I®M « E°¯
± sJp³² ¦µ´ £
¦§ ³¦ ¨m¦©´¶ Ÿ¸· ¹
´™¿ ¡¼½ ´¦µÀ˜·;Ÿ¸· ¹
~{|Âpuq>w™piprqšsosJ’›}sJ—˜Ã‘w™Ä6sJ~fw‘tus
ºO´
Oº ´
¡ ¨˜»@ºO´¦¼ }» ¡¼½ ·9Ÿ¾·9¹
¡ }»@ºO´¦¼Á}»
Ÿ¸·
™´ Å
ŸÆÁÇ´ ¿ ŸÈ¦§ w‘— ± puq6sm|}tu’U›}’—’U~fw‘—Ä6—6~kpw‘z6Usm~rw ±6± sm–+|‘’—˜pKÉ
Ê
Ë Ì„Í
·ωÐJÑ4Ò ÓoËÔQËÔÕ
dÖ ×Ø‘ÙuÙ{ÚU۔ÜxÝuÞ6ßmà@á{ÚUÝrÚUàKؑ×+â+ã‘ڐä˜ÝؙÝiÝrޚßoã‘áuڐå‘ڐäxÛOã}áiÝuÞ6ßmÙ{ÜSÙ{Ýuß@æ
ç˜è
ç˜é ê ë è¢ìîí™ï
ç˜ï
ç}é ê ð í!è ð í!ï
ñ¬ò&ó#ô\õ®öOò+÷°ø
í
ù ßJݳú 4ë ðµ
ì¤í ÿOûð ë ,ì}í ê¾ý9þ
û
íð
íð ©
¸
ê
ý
þ
O
ÿ
û
ð ûü
ì
ì
û û ý;ê¸ý þ ÿOû ì
ì ê ý
Ù{ãÂÝuÞ>ؙÝiÝrޚßoßJڐå}ßJä ‘Ø™× 6ßJÙfؑáuß
û°ê ð ì ëfû êÈð ð ë
ؑä ù ÝuÞ6ßmã}áuÚUå}ڐäÚUÙfØ Ù{ÝØ 6אßmÙuâšÚárØ™× Ø‘×U׬Ù{ã}× šÝuڐã}äšÙ Ýrß@ä ù ÝuãJß@á{ãxؑ٠é !#"$&%('*),+
-/.102.1023
4657898;:=<?>A@9BDCFEHG;:=@I:=EJ75LKLM:N@O7@P@IBQC/MG9:R:NA<SMGP@9BDCF8;>T8;@9CHU
VW
VX
Va`
VX
Y
Z\[
W
Y
W^]_`
Zcb
`
dfehgjilk
mSeLnpo
q C*@sr Zs[/Zct
u
_
Yy
ZOb/Zvtxw
] u
]€ u
t
yt
Y y
z
{|t
z
{|t
]
[}H{St
]
}H{St
]
ba}~Z
]‚_
_
Yy/z
}Yy
8;M@9Bƒ7@P@IBQC/C*:RC*N„75…DC*87G9C
t†‡YˆZ
_Š‰
t
Y‹Z


7N q @9BDCPMG;:R:=NŒ:8p78Ž@7Q5CP:UKDG;MKLC*G~NDM q C‘’75=5“8;M5…Q@9:MNQ8~@IC*N q @IMF”HC*G9M/78 Xp•
_
–—
˜
™#š›&œ(*ž,Ÿ
/¡1¢2¡1¢¤£
¥~¦=§D¨Š©§D¨«ªH¬©­9­;¦=®?¯&°I±D²FªH³;¦=°9¦ªJ©¬f´hµ¦§°µ®¶°I±D²F¬=¦§D²H©³·­;¯T­Ž°I²H¸
¹º
¹»
¹Á
¹»
¼
º^¿ÀÁ,¿Â
½s¾
º
¼
ÀÁ¿Ã
½
Ä ÅhÆjÇlÈ
ÉSÅLÊpË
f
¥~¦=³9­Ž°·Ì6²/­9²¤°°I±D²F³9±Q­O²*Ía΃©¬l°Iµ^ÏH²*³9µ^°Iµ^¨D²¤°I²H³;¸&¦=§D²°I±Q²/ª*³9¦Ð°I¦ªH©¬Ñ´hµ¦§°HÒ
½O¾
º
ºÓ¿ÀÁ
ÀÁ
½
¼
Â
½
¼
½
ÃcÔ
Ú µQÛQ°I±D²Fª*³9¦=°9¦ªH©¬Ñ´hµ¦§°P¦=­/Ü ºÞÝIÁQß
æ ¼
º
½\Ö
ÀÁ
Õ
¼
º
Õ
¼×½sØ
ºÙ¿Ã
Õ
Á
¼
Ã
ÂÔ
¼
Ü ÃÝJÃ
ߤàâá ²F¦§°9³9µT¨DÎDªH²,㩳;¦ä©åD¬=²H­
¼
º
ÃÝç
½
Á
¼
Ãè
½
é꧊°I²*³9¸­Pµ®PÜ æ IÝ ç“ß °I±D²F­;¯T­Ž°I²H¸ëåh²Hª*µ¸²H­
¹ æ
¹»
¹aç
¹»
¼
¿Àç
Ü ¿Ã
ßl¿À Ü ç¿Ã
ßl¿‚Â
½O¾ æ
¼‹½s¾ æ
¼
À Ü ç\¿Ã
ßl¿Ã
Ü æ ¿Ã
ß
½
¼
æ ½ Àç
á ²,샧D¨«§DµíÌ°I±Q²/²*¦î²*§ã©¬ÎD²*­HÒ
¨D²*°sï ½O¾ð½cñ
Ã
À
½
À
½vñxò
¿‚õ ¿‚õ
ñô
ñ
¼ ó
¿Àß Ü
Ü
Õ
¼ó
Õ
Ü
ñ
ñ
¿
ñ
¿
¾
ß
½
À
¼ó Õ
ß
Ö p
ô ½cÖð¼öó
­;µ°9±ƒ©°P°I±Q²/²*¦î²*§ã©¬ÎD²*­©³9²
ø
¼×½\Ö,½
Ö
ô
©§Q¨°I±Q²x´hµ¦§°úÜ Ã Û Ã
ß ¦=­&©û­;°I©åD¬²Š¦=¸&´D³;µ´h²H³Ó§DµT¨D²Òú©¬¬P­;µ¬Î“°I¦µ§D­Ó°I²H§Q¨°Iµü¦=°©­
»ý þà
ñ÷¼‹½sÖ
¿ùø
Ý
Ö
¾
ñ
"!#$%$&')(*,+-."/&*%#01!32546#7*84'9*,+-.!#-"/:$&(;$<*,-"=
>6?
>> 6@ A BDC ? BFEGBIH
>6E @ A JK ?ML E L C6N
O3P5QSRUTWVXPZY\[
/%$<*:]^-$%-_*8*,+-./%+`$a-
bdce!U*,4gf"-
/%4g*,4g-_*,-"/&=-h*,+`-
/%i*,#"!j2546#7*"k
n ?
m
n
?
?
BlC ?ML BIE A H
Jo ? ApBDJo
ApBGq m
JK E A BDC6N
EgArBlC BFHsApBq
t 4`u`*,+-.
/%*%#"!j2546#7*v$w ?yx E`zA w Bq x BqWz{}| -.#7*%/%4;c"-~1/&€!-"$
 A ?ML q x8‚ AƒE L qg„
…†*,-
/%=‡$v4'vw  x,‚ z *,+-.$&(;$<*,-"=ˆ€5-"
46=‡-"$
>
>>d7‚@ A BlC w  B‰qWzŠB w ‚ B‰qWzŠBIHGArBlC  B ‚
>6@ A JK w  B‰q‹z L w ‚ BŒqWz L C6NAJK  LŽ‚
| -e 4‹]*,+`--
#‘6-
7~1!#c-
$"k
-
*l’ BlC.BI“ Bq AƒN n w “DB‰qWz w “ L C7z L JKsA•N n
JK L qvBIL “ ”
n w L L
“—– J1“ JoGAƒN
“ qWz<– J6˜.AN
$&4‡*%+e1*v*,+`--
#‘6-
7~1!#c-
$8/%“1™\ApBq L ˜1š x “ – ArBGq8BŽ˜1š
`*%+-.2Z4#7*Gw)› q uœ› qWz $8‡$<*€!-$%2/,!Sk!!j$%46!c`*,46$^*%-"*,4gi*a$ @\ ž {
ÿ
˜
Ÿ ¡¢£¤
¥¦ §¨©¨©‹§ª
^« ¬#­®%®&¯°)±²,³`´sµ"¶&¯²%¯#µ0­1¬9·5¸6¯¹d²D­1²a²,³´G¸6¶%¯º6¯#¹°X¸6¶8²,³´s®&±;®&²%´"»¼­¹½¾®&¿´
²%µ³²,³´M·`³e­®%´
·¬#­¹´½¯#­º6¶%­»
À6Á ÁÄÆÅÇ
À7À6Ç à ÁĎÉ1Ç
À6Â Ã È
Ê3Ë5ÌSÍUÎWÏXËZÐ\Ñ
Ò ´Óe¹½²,³´.´"¯º6´"ÄF¹7Ô1Ù ­¬ÕÄl´"®
Å Ö
ÙvÞßÉ6à Ý ÙhÄ àjÞ ÈGÕÛsÜ
D
Ä
É
I
Ä
Ù
Ú
ƒ
Ã
Û
Ü
Ý
½´
²l×IÈØ Ù1áÞßÅ1ÙvÞãâ
ÃƒÛ Ü Ý ÙvÞßâ6à Ý Ù8Þ Ø à ÃØÛ
Ø
®&¸‡²%³e­1²v²,³`´´
¯#º6´
¹7Ô­1¬#Õ´
®8­¶%´ Ù1ä Ã Ä Ù á Ã Ähâ
؇å
­¹`½²%³´.¸6¶%¯º6¯#¹¯®8­¹Õ¹®<²­æ¬´.®,­½½`¬#´.à ·Z¸Ù ¯#¹7²0çèa¸‹é²,³´.´
¯#º6´
¹dÔ´"µ_²,¸6¶&®"Ö
êëÍUì ì)ÍUì Ö
í
Ä
Ù
ÄlÅ Á
¸6¶
Û Úïî
Ú
Ú
h
Ä
h
É
I
Ä
Ù
Ç
Ã
×
× Û
Ù1ä Ã×ÆØÄ È
Ø çlð¯º6´"¹7Ô1­¬#Õ`´
Ø Ö È Á Ä Å1Ç Ã Û
È Á Ä Å1Ç Ã Û îå
ñ ³`´®&¸6¬#Õò²,¯#¸¹¯#®
äÃ Ú î
Í ×ôó
Ù
h
Ä
â
Ø
È çlð¯º6´"¹7Ô1­¬#Õ`´ á à Ö
Å1Á Ä Å1Ç Ã Û
È Á Ä È Ç Ã Û îå
ñ ³`´®&¸6¬#Õò²,¯#¸¹¯#®
Í á Ã ×ŽØ Ú î
Ø
õ òö ÷`øWù`ú7ö7ù^û
õ ü5ý‹ù7þ ÿ ù;øŠû ü5ý‹ù7þ ÿ
—ù øŠû ü5ýWù7þ ÿ—ùdù;øŠû
"! #$%$!'&(&'%)*+$*-,./%.)%0 .)
*1'(%&'234)5.#6*'7.'$89$,."%.)%0 .)
:;=<?>6@BADC EF(GIH6JKC.LMNCOF(PQCRJKSS%T@K;6UVF<WEYX[Z]\%L^\_X`ZbacEed
f'g]\6hiLMj@B<?>k<MNSlAQC<mAbS(no@K;@B<?@KF(J
p S%;6q6@r<?@KS(;6A4sjdtEedvu$hiLw\Ydvu%hmhmZVd
f%L[u%h LxdyEedzu$h L^\Ydvu$hQhmZVdc{L|u$hi}
'~
€'‚
Y€ 
_ ƒ„ '…… Q† _‡ _$† _(ˆ‰6Š.‹ Œx‡
„ …… ‹ _‡ _ŽŠ† Y.ˆ$6Š(‹ Y'‘‡
ƒ’† _‡
‹ _'‘‡ ˆ“‘””“•‡ – .— ˜ƒ’† Œ‡
‹ Œ'‘‡
ƒƒ’† m‚6_€™‡
‹ m‚6.‚%‘‡ƒ† m‚6ˆ%š˜‡
‹ ?‚6‚$‘‡ƒ† m‚6.‚•‡
‹ m‚6Y€#‘‡ƒ
† m‚6($š‘‡Q‹ m‚6(‚%N‡
†ˆ$šN””’š‘‡›‹ˆ%šN””š‘‡œ #— – Nƒ %„‘‡'ž •‡~
.—W‡’ –'Ÿ‘‡
6 . .¡¢££˜‡
.„ Ÿ…š“•‡
. '¥ $‚˜”
‚“6
¤
„ %‹ i¦# Y€(§‡
$Y¨©
ª©
™¨Œ‡
%…'— «¬ƒ’­ £$® ¯_°± ¢‘‡
²³ £ ‡/€'Ž%‘‡ . ´µƒ¨†™¨Œ‡#¨‹e¨4‘‡¶ . .…6'— ·¸ƒc­ £$® ¯_°.± ¢‘‡ ²³ £ ‡5€‚%¹
õ ý ý‹ù5øˆû \û
õ —ù\—ù ÿ ý û ÿ ÿ ù ù ÿ<ù ÿ ù ý ÿ \ÿ<ù <ù ÿ ù ÿ ù
\ÿ<ù ÿ ÿ<ù ÿ ù ÿ íø ÷ \÷ ÿ ÿ ù \ÿ ù ÿ
ÿ 6ú Æú6ÿ ö;ö —ü ø ÿ
ò
ý
÷
1
ö
ò
ö
÷
÷
d
ò
ö
d
÷
÷
÷ùdþ 7ö ÆøWù`÷ 5øòý ò÷ Šû
õ ý7ø 7ú ÿ 6ú ÷—ù;ø ¾ù5ýWù 6÷ 7þú`ø÷6ú ÷1ö7ù;ö`;úù ý‹ù ¾ùZý‹ù 6÷ ;ù
Phase Portrait
4
y
–4
–2
2
0
2
4
x
–2
–4
Legend
ºS%PjFH6JBS(<DS(nEed»<ihWS%;JB\%}r}r}B}
Curve
Curve
Curve
Curve
Curve
1
2
3
4
5
¼
½¿¾ÁÀl›ÃÄRÅ
ÆÈÇkÉiÇkÉÆËÊ
Ì ÍKÏ(ÐmÐQÑBÒyÓkÔ?ÕÖO×4ØQÑBÔmÑK×#Ï.ÍxÙ_Ú%ÑBÛÔ]Ï.ÔÜÔ?Õ6ÖËÚ%ØmÑBÝ%ÑKÛ=Ò»Ú%ØjÔ?Õ6ÖOÐQÓÐQÔmÖ4ÞµÏ(Û6ß8ÐQà(ÖÔm×á՛Ô?Õ6ÖâÙÕÏ(ÐmÖ
Î
Ù6ÍKÏ(Û6ÖRß6ÑKÏ(Ý%ØmÏ(Þ
ã%ä
ã%å
ã$ê
ã%å
äâçéè.ê
æ
æ
älì"è.ê
ë
íYî_ïvð™ñ'ò»îŒó‘ô
õ ÖRöÛ6ß=Ô?Õ6Ö÷Ö4ÑBÝ%Ö4Û$ø.Ï(ÍBù6Ö4Ðú
ß6Ö Ôbû/ü
ë
ìþý
è
ì]è÷ì¶ý ÿ æ
ýoç"ý ì è
ü æ
ÐQÚIÔmÕÏ.ÔDÔ?ÕÖÈÖÑKÝ%ÖÛ$ø(Ï.ÍKù6ÖÐjÏ(ØmÖ
ýjç3è ý]ì
ýjç
ý]ì
ì
ü
ü
æ
æ
ý
Üý
ì
æ
æ
Ï(Ûß=ÔmÕ6Ö÷Ú%ØmÑBÝ%ÑKÛkÑBÐjÏ(Û=ù6Û6ЍÔáÏ6ÍBÖ÷Ð?Ï(ß6ßÍKÖ÷ٌÚ(ÑKÛ$ÔÜÚªÔ?Õ6Ö÷ÖÑKÝ%ÖÛø(Ö4× Ô?Ú%ØQÐ4ú
kð
Ú%Ø
ì¶ý
û/ü
ü
"!ÁÑBÝ%Ö4Û$ø.Ï(ÍKùÖ ý
æ
ú
ë
è
ä
û
ì è ìþý ÿ
ê ÿ æ
ì èä
ç
ä
ë
# ÕÖÈÐQÚ%ÍKùÔ?ÑKÚ(ÛkÑKÐ
ð$
è "!ÁÑBÝ%Ö4Û$ø.Ï(ÍKùÖ ý
æ
è.ê
ì
ë
ä
ë
ä
ç
èê
è ê
ð
ç
æ
û
ÿ
ü
æ
ì
ë
æ
è
ÿ
æ
û"ü
æ
û
æ
ê
ì" ú
ë
# ÕÖÈÐQÚ%ÍKùÔ?ÑKÚ(ÛkÑKÐ
ýyð ú
æ
ÿ
%
%
& '*),+*-&+/.
(
021+354768+:99<;<)<=>.
021+354@?A;9+A)(=>.
021,+354B?A;<9++:99<;<)<=>.
C<DEF*GHFIKJMLNOPF(DRQ*OTSRN(DFIOS,UU,EV<WXN(DWZY<OOF
EOVO[<GRF<OV
C<DEF*GHFIKJMLNOPF(DRQ*O<W\D,EEU,]^DFV_LEDF`W,aD,LObN(DcOPY<OOFTEOVO[<GRF<OV
dBegfih`jlknmRo*pqsr`tumv*wxmypz{m|tu}<}~jue`€pfoƒ‚…„"o`†ˆ‡,‰vŠ‰‹‚…„Œ,oŽ7‡‰v:wjlfih`‘mk{mHf’k"}“
jle`jlf’j”pt2•}e*–`jlf’ju}e`kH—
˜ o š˜ ™› Šv ‰ œ˜ ™›{› „ 7˜  v ™› v
˜ o ˜š™› Šv ‰ œ˜ ™›{› „ ˜ ‡<v ™› v ˜ o ˜œ™› vž‰ ˜œ™›’› „ ˜ Ž  v ™› v ˜ o ˜š™› vŠ‰ ˜œ™›{› „ ˜œ™ v ,› v ˜ o ˜š™› vŠ‰ ˜œ™›{› „ ˜œ™ v<Ž7‡ ›RŸ
%
621, A1+2)¡.B¢¤£,¥¦.
?A;9+ƒ£§.B¢
%
6 8?2;9+¦4*¨B©A1,ªª4{«4@+2=K¬@+2=¢<­*®«47+2=¯°4@+‹=±¬@©A1ª<ª47°¦47+2=K¬7+2=¢<²:®«4@+ƒ=³­*®°4@+ƒ=,´¦¬
¨µ«47+2=K¬@°47+2=,´¦¬¶+*¢<³²¦··7²$¬¸)¹'º`'¢5¨µ«4@+‹=K¬@°4@+‹=,´¦¬
¨¨µ«¦4’¥`=¢2£¬@°4’¥`=¢<¥´¦¬(¨B«4’¥`=¢(­$¬7°4i¥:=¢<¥´5¬(¨µ«¦4i¥`=»¢¼³2£½¬@°4’¥:=¢<¥´¦¬*¨µ«4’¥`=¼¢
¥$¬@°¦4’¥`=¢2£´¦¬(¨µ«4i¥`=¢<¥$¬7°4’¥`=¢(³­´´5¬¸«(¢<³¾x··B¾¦¬b°<¢<³¾x·<·µ¾¦¬
;(1º:'*¹9;9&<¢5¨µ&('©¦¬µ¿A;,À:'$¬@ &('<'º/¬ÂÁA- ('º(+`-$¬µ¿2;-*¹,Ã(´5¬Ä-&&`902)¢<ÅÆ(Ç<ÈȦ¬ZÁ:'+3A9©<¢&Ã<ª¾(²¬
),+*'?2)<1,É('¢<¥¦·@¥²:=>.
%
©A1)?A;-°4ËÊ?A;<9+ƒ£ÌK¬b+21,+:;'¢ƒÍHÎ3`-*)'»Î:9&+<&*-`1+͋¬b+‹1+:;'ª:9,º<+ϢШµÑ8<ÈÒ8<Ó2ÔÕǦ¬
Ö(× È65¬
£,­´¦¬P;-¿`'*;<)آ٨HÍH«Í‹¬ÍH°½Í´¦¬¸;-¿`'*;ª`9,º<+Ú¢Û¨ÂÑ8(ÈÒ8Ó2ÔÕǦ¬ Ö(× È65¬_£¥´A=KÜ
Phase Portrait
4
y
–4
–2
2
0
2
4
x
–2
–4
Legend
Curve
Curve
Curve
Curve
Curve
Curve
1
2
3
4
5
6
Ý:ÞߏàMá`âlÞãnÞäKå½æçãËèÞé*âlêëìëìëlë
í
îAïðñ*òÐóBô
õö î2ïðñ¦÷*øBùAú,ûû÷{ü÷@ñ2ýKþ@ñ2ýô<ò*ÿü÷7ñ2ý÷@ñ‹ý±þ@ùAúû<û÷¦÷7ñ2ýKþ7ñ2ýô:ÿü÷@ñƒýò*ÿ÷@ñƒý
¦þ
øµü÷7ñ2ýKþ÷7ñ2ý
¦þ¶ñ*ô$þô5øÂñ5þ@ü¦÷@ñ2ý
¦þ
øøµü¦÷`ýôþ÷`ýô¦þ(øBü÷`ýô(ò$þ÷:ýô5þ(øµü¦÷`ý»ô½þ÷:ýô¦þ*øµü÷`ý¼ô
$þ¦÷`ýô¦þ(øµü÷`ýô$þ÷`ýôò5þ¸ü(ô ¦þ!<ô"¦þ
ï(ú#$%ðïð&<ô5ø"&ù¦þ"'Aï
($$þ)&/þ+*-,)(ñ,$þ"'2ï,%
.5þ/,&&`ð10ô23455¦þ6*$ñ7Aðù<ô&.<ûþ
,ñ%î<ú
8ô:ý>ó
í
ùAúîAï,÷:9îAï<ðñ*ò%;Kþbñ2ú,ñ:ïô=<2(ðï
(<ñ‹úð
?>(%&@%A<‹þbñ‹úñ:ïû:ð
<ñÏôÐø"B ö 5C öDE >4¦þ
FG 5 õ þH,ò¦þPï,'*ïØôÙøI<RñJ<‹þ#<HüK<L¦þ¸ï,'*ïû`ð
<ñÚôÛø+B ö 5C öDE >4¦þ FG 5 õ þM1AýJN
Solution Curve
4
x
–4
–2
2
0
2
4
t
–2
–4
Legend
Curve
Curve
Curve
Curve
Curve
2
3
4
5
6