Systems Dynamics Modelica by Example.pdf

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h?2 #BHBiv iQ F22T 2[miBQMb BM i?2B` ?i2ti#QQF 7Q`K? Bb BKTQ`iMi #2+mb2- b r2 rBHH b?Qr BM Hi2`
b2+iBQMb- r2 2p2MimHHv rMi iQ ;2i iQ i?2 TQBMi r?2`2 i?2b2 2[miBQMb `2 ?+Tim`2/? BM BM/BpB/mH
+QKTQM2Mib KQ/2HbX AM i?Qb2 +b2b- r2 rQM?i FMQr Ur?2M r2 +`2i2 i?2 +QKTQM2Mi KQ/2HV 2t+iHv r?i
p`B#H2 2+? 2[miBQM rBHH #2 mb2/ iQ bQHp2 7Q`X JFBM; bm+? KMBTmHiBQMb i?2 `2bTQMbB#BHBiv Q7 i?2
JQ/2HB+ +QKTBH2` MQi QMHv KF2b i?2 KQ/2H /2p2HQTK2Mi 7bi2` M/ 2bB2`- #mi Bi /`KiB+HHv BKT`Qp2b
i?2`2mb#BHBivQ7 i?2 KQ/2HbX
h?2 7QHHQrBM; };m`2 b?Qrb i?2 /vMKB+ `2bTQMb2 Q7 i?2_G*RKQ/2H,
Rk *?Ti2` RX .2b+`B#BM; "2?pBQ`

23. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
1tTM/BM; QM i?2b2 2H2+i`B+H 2tKTH2b
b K2MiBQM2/ BM i?2S`27+2UT;2 AV- i?2 bi`m+im`2 Q7 i?Bb #QQF HHQrb mb iQ 2tTHQ`2 KQ`2 ?vT2`K2/B
#b2/ TT`Q+? BM r?B+? `2/2`b `2 2M+Qm`;2/ iQ T`Q+2bb i?2 Ki2`BH i?i Bb KQbi HB;M2/ rBi?
i?2B` ;QHb M/ BMi2`2bibX h?2 M2ti +?Ti2` rBHH T`2b2Mi KQ/2H r?Qb2 2[miBQMb `2 /2`Bp2/ 7`QK
K2+?MB+H bvbi2KX A7 vQm rQmH/ T`272` BMbi2/ iQ b22 i?Bb 2H2+i`B+H 2tKTH2 2ti2M/2/ iQ BM+Hm/2 KQ`2
+QKTH2t #2?pBQ`- vQm Kv rMi iQ bFBT ?2/ iQ i?2arBi+?2/ _G* *B`+mBiUT;2 eRV 2tKTH2X
J2+?MB+H 1tKTH2
A7 vQm `2 KQ`2 7KBHB` rBi? K2+?MB+H bvbi2Kb- i?Bb 2tKTH2 KB;?i ?2HT `2BM7Q`+2 bQK2 Q7 i?2 +QM+2Tib
r2?p2 +Qp2`2/ bQ 7`X h?2 bvbi2K r2 rBb? iQ KQ/2H Bb i?2 QM2 b?QrM BM i?2 7QHHQrBM; };m`2,
RXRX "bB+ 1[miBQMb Rj

24. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
Ai Bb rQ`i? TQBMiBM; Qmi ?Qr Km+? 2bB2` Bi Bb iQ +QMp2v i?2 BMi2MiBQM Q7 KQ/2H #v T`2b2MiBM; Bi BM
b+?2KiB+ 7Q`KX bbmKBM; TT`QT`Bi2 ;`T?B+H `2T`2b2MiiBQMb `2 mb2/- 2tT2`ib +M p2`v [mB+FHv
mM/2`biM/ i?2 +QKTQbBiBQM Q7 i?2 bvbi2K M/ /2p2HQT M BMimBiBQM #Qmi ?Qr Bi Bb HBF2Hv iQ #2?p2X
q?BH2 r2 `2 +m``2MiHv 7Q+mbBM; QM 2[miBQMb M/ p`B#H2b- r2 rBHH 2p2MimHHv rQ`F Qm` rv mT iQ M
TT`Q+? UBM i?2 mT+QKBM; b2+iBQM Q7 i?2 #QQF QM*QKTQM2MibUT;2 R3jVV r?2`2KQ/2Hb rBHH #2 #mBHi
BM i?Bb b+?2KiB+ 7Q`K 7`QK i?2 #2;BMMBM;X
6Q` MQr- ?Qr2p2`- r2 rBHH 7Q+mb QM ?Qr iQ 2tT`2bb i?2 2[miBQMb bbQ+Bi2/ rBi? i?Bb bBKTH2 K2+?MB+H
bvbi2KX 1+? BM2`iB ?b `QiiBQMH TQbBiBQM-ϕ- M/ `QiiBQMH bT22/-ωr?2`2ω=˙ϕX 6Q` 2+?
BM2`iB- i?2 #HM+2 Q7 M;mH` KQK2MimK 7Q` i?2 BM2`iB +M #2 2tT`2bb2/ b,
J˙ω=
!
i
τi
AM Qi?2` rQ`/b- i?2 bmK Q7 i?2 iQ`[m2b-τ- TTHB2/ iQ i?2 BM2`iB b?QmH/ #2 2[mH iQ i?2 T`Q/m+i Q7 i?2
KQK2Mi Q7 BM2`iB-J- M/ i?2 M;mH` ++2H2`iBQM-˙ωX
i i?Bb TQBMi- HH r2 `2 KBbbBM; `2 i?2 iQ`[m2 pHm2b-τiX 6`QK i?2 T`2pBQmb };m`2- r2 +M b22 i?i i?2`2
`2 irQ bT`BM;b M/ irQ /KT2`bX 6Q` i?2 bT`BM;b- r2 +M mb2 >QQF2?b Hr iQ 2tT`2bb i?2 `2HiBQMb?BT
#2ir22M iQ`[m2 M/ M;mH` /BbTH+2K2Mi b 7QHHQrb,
τ=c∆ϕ
6Q` 2+? /KT2`- r2 2tT`2bb i?2 `2HiBQMb?BT #2ir22M iQ`[m2 M/ `2HiBp2 M;mH` p2HQ+Biv b,
τ=d∆˙ϕ
A7 r2 TmHH iQ;2i?2` HH Q7 i?2b2 `2HiBQMb- r2 ;2i i?2 7QHHQrBM; bvbi2K Q7 2[miBQMb,
ω1=˙ϕ1
J1˙ω1=c1(ϕ2−ϕ1)+d1
/(ϕ2−ϕ1)
/t
ω2=˙ϕ2
J2˙ω2=c1(ϕ1−ϕ2)+d1
/(ϕ1−ϕ2)
/t
−c2ϕ2−d2˙ϕ2
G2i?b bbmK2 Qm` bvbi2K ?b i?2 7QHHQrBM; BMBiBH +QM/BiBQMb b r2HH,
ϕ1=0
ω1=0
ϕ2=1
ω2=0
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KQpBM; UBX2X-ω=0V- #mi i?2`2 Bb MQM@x2`Q /2~2+iBQM +`Qbb #Qi? bT`BM;bX
SmHHBM; HH Q7 i?2b2 p`B#H2b M/ 2[miBQMb iQ;2i?2`- r2 +M 2tT`2bb i?Bb T`Q#H2K BM JQ/2HB+ b 7QHHQrb,
KQ/2Ha2+QM/P`/2`avbi2K] b2+QM/ Q`/2` `QiiBQMH bvbi2K]
ivT2M;H24_2HUmMBi4]`/]Vc
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ivT2AM2`iB4_2HUmMBi4]F;XKk]Vc
ivT2aiB77M2bb4_2HUmMBi4]LXKf`/]Vc
ivT2.KTBM;4_2HUmMBi4]LXKXbf`/]Vc
T`K2i2`AM2`iB CR4yX9]JQK2Mi Q7 BM2`iB 7Q` BM2`iB R]c
T`K2i2`AM2`iB Ck4RXy]JQK2Mi Q7 BM2`iB 7Q` BM2`iB k]c
T`K2i2`aiB77M2bb +R4RR]aT`BM; +QMbiMi 7Q` bT`BM; R]c
T`K2i2`aiB77M2bb +k48]aT`BM; +QMbiMi 7Q` bT`BM; k]c
T`K2i2`.KTBM; /R4yXk].KTBM; 7Q` /KT2` R]c
T`K2i2`.KTBM; /k4RXy].KTBM; 7Q` /KT2` k]c
M;H2 T?BR]M;H2 7Q` BM2`iB R]c
R9 *?Ti2` RX .2b+`B#BM; "2?pBQ`

25. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
M;H2 T?Bk]M;H2 7Q` BM2`iB k]c
M;mH`o2HQ+Biv QK2;R]o2HQ+Biv Q7 BM2`iB R]c
M;mH`o2HQ+Biv QK2;k]o2HQ+Biv Q7 BM2`iB k]c
BMBiBH2[miBQM
T?BR4yc
T?Bk4Rc
QK2;R4yc
QK2;k4yc
2[miBQM
ff 1[miBQMb 7Q` BM2`iB R
QK2;R4/2`UT?BRVc
CR /2`UQK2;RV4+R UT?Bk@T?BRVY/R /2`UT?Bk@T?BRVc
ff 1[miBQMb 7Q` BM2`iB k
QK2;k4/2`UT?BkVc
Ck /2`UQK2;kV4+R UT?BR@T?BkVY/R /2`UT?BR@T?BkV@+k T?Bk@/k /2`UT?BkVc
2M/a2+QM/P`/2`avbi2Kc
b r2 /B/ rBi? i?2 HQr@Tbb }Hi2` 2tKTH2-_G*R- H2i?b rHF i?`Qm;? i?Bb HBM2 #v HBM2X
b mbmH- r2 bi`i rBi? i?2 MK2 Q7 i?2 KQ/2H,
KQ/2Ha2+QM/P`/2`avbi2K] b2+QM/ Q`/2` `QiiBQMH bvbi2K]
L2ti- r2 BMi`Q/m+2 T?vbB+H ivT2b 7Q` `QiiBQMH K2+?MB+H bvbi2K- MK2Hv,
ivT2M;H24_2HUmMBi4]`/]Vc
ivT2M;mH`o2HQ+Biv4_2HUmMBi4]`/fb]Vc
ivT2AM2`iB4_2HUmMBi4]F;XKk]Vc
ivT2aiB77M2bb4_2HUmMBi4]LXKf`/]Vc
ivT2.KTBM;4_2HUmMBi4]LXKXbf`/]Vc
h?2M r2 /2}M2 i?2 p`BQmb T`K2i2`b mb2/ iQ `2T`2b2Mi i?2 /Bz2`2Mi T?vbB+H +?`+i2`BbiB+b Q7 Qm`
bvbi2K,
T`K2i2`AM2`iB CR4yX9]JQK2Mi Q7 BM2`iB 7Q` BM2`iB R]c
T`K2i2`AM2`iB Ck4RXy]JQK2Mi Q7 BM2`iB 7Q` BM2`iB k]c
T`K2i2`aiB77M2bb +R4RR]aT`BM; +QMbiMi 7Q` bT`BM; R]c
T`K2i2`aiB77M2bb +k48]aT`BM; +QMbiMi 7Q` bT`BM; k]c
T`K2i2`.KTBM; /R4yXk].KTBM; 7Q` /KT2` R]c
T`K2i2`.KTBM; /k4RXy].KTBM; 7Q` /KT2` k]c
6Q` i?Bb bvbi2K- i?2`2 `2 7Qm` MQM@T`K2i2`p`B#H2bX h?2b2 `2 /2}M2/ b 7QHHQrb,
M;H2 T?BR]M;H2 7Q` BM2`iB R]c
M;H2 T?Bk]M;H2 7Q` BM2`iB k]c
M;mH`o2HQ+Biv QK2;R]o2HQ+Biv Q7 BM2`iB R]c
M;mH`o2HQ+Biv QK2;k]o2HQ+Biv Q7 BM2`iB k]c
h?2 BMBiBH +QM/BiBQMb Ur?B+? r2 rBHH `2pBbBi b?Q`iHvV `2 i?2M /2}M2/ rBi?,
BMBiBH2[miBQM
T?BR4yc
T?Bk4Rc
QK2;R4yc
QK2;k4yc
h?2M +QK2 i?2 2[miBQMb /2b+`B#BM; i?2 /vMKB+ `2bTQMb2 Q7 Qm` bvbi2K,
2[miBQM
ff 1[miBQMb 7Q` BM2`iB R
QK2;R4/2`UT?BRVc
CR /2`UQK2;RV4+R UT?Bk@T?BRVY/R /2`UT?Bk@T?BRVc
ff 1[miBQMb 7Q` BM2`iB k
RXRX "bB+ 1[miBQMb R8

26. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
QK2;k4/2`UT?BkVc
Ck /2`UQK2;kV4+R UT?BR@T?BkVY/R /2`UT?BR@T?BkV@+k T?Bk@/k /2`UT?BkVc
M/ }MHHv- r2 ?p2 i?2 +HQbBM; Q7 Qm` KQ/2H /2}MBiBQMX
2M/a2+QM/P`/2`avbi2Kc
h?2 QMHv /`r#+F Q7 i?Bb KQ/2H Bb i?i HH Q7 Qm` BMBiBH +QM/BiBQMb ?p2 #22M ??`/@+Q/2/? BMiQ i?2
KQ/2HX h?Bb K2Mb i?i r2 rBHH #2 mM#H2 iQ bT2+B7v Mv Hi2`MiBp2 b2i Q7 BMBiBH +QM/BiBQMb 7Q` i?Bb
KQ/2HX q2 +M Qp2`+QK2 i?Bb Bbbm2- b r2 /B/ rBi? Qm`L2riQM +QQHBM; 2tKTH2bUT;2 eV- #v /2}MBM;
T`K2i2`p`B#H2b iQ `2T`2b2Mi i?2 BMBiBH +QM/BiBQMb b 7QHHQrb,
KQ/2Ha2+QM/P`/2`avbi2KAMBiS`Kb
] b2+QM/ Q`/2` `QiiBQMH bvbi2K rBi? BMBiBHBxiBQM T`K2i2`b]
ivT2M;H24_2HUmMBi4]`/]Vc
ivT2M;mH`o2HQ+Biv4_2HUmMBi4]`/fb]Vc
ivT2AM2`iB4_2HUmMBi4]F;XKk]Vc
ivT2aiB77M2bb4_2HUmMBi4]LXKf`/]Vc
ivT2.KTBM;4_2HUmMBi4]LXKXbf`/]Vc
T`K2i2`M;H2 T?BRnBMBi4yc
T`K2i2`M;H2 T?BknBMBi4Rc
T`K2i2`M;mH`o2HQ+Biv QK2;RnBMBi4yc
T`K2i2`M;mH`o2HQ+Biv QK2;knBMBi4yc
T`K2i2`AM2`iB CR4yX9]JQK2Mi Q7 BM2`iB 7Q` BM2`iB R]c
T`K2i2`AM2`iB Ck4RXy]JQK2Mi Q7 BM2`iB 7Q` BM2`iB k]c
T`K2i2`aiB77M2bb +R4RR]aT`BM; +QMbiMi 7Q` bT`BM; R]c
T`K2i2`aiB77M2bb +k48]aT`BM; +QMbiMi 7Q` bT`BM; k]c
T`K2i2`.KTBM; /R4yXk].KTBM; 7Q` /KT2` R]c
T`K2i2`.KTBM; /k4RXy].KTBM; 7Q` /KT2` k]c
M;H2 T?BR]M;H2 7Q` BM2`iB R]c
M;H2 T?Bk]M;H2 7Q` BM2`iB k]c
M;mH`o2HQ+Biv QK2;R]o2HQ+Biv Q7 BM2`iB R]c
M;mH`o2HQ+Biv QK2;k]o2HQ+Biv Q7 BM2`iB k]c
BMBiBH2[miBQM
T?BR4T?BRnBMBic
T?Bk4T?BknBMBic
QK2;R4QK2;RnBMBic
QK2;k4QK2;knBMBic
2[miBQM
QK2;R4/2`UT?BRVc
QK2;k4/2`UT?BkVc
CR /2`UQK2;RV4+R UT?Bk@T?BRVY/R /2`UT?Bk@T?BRVc
Ck /2`UQK2;kV4+R UT?BR@T?BkVY/R /2`UT?BR@T?BkV@+k T?Bk@/k /2`UT?BkVc
2M/a2+QM/P`/2`avbi2KAMBiS`Kbc
AM i?Bb rv- i?2 T`K2i2` pHm2b +M #2 +?M;2/ 2Bi?2` BM i?2 bBKmHiBQM 2MpB`QMK2Mi Ur?2`2 T`K2i2`b
`2 ivTB+HHv 2/Bi#H2 #v i?2 mb2`V Q`- b r2 rBHH b22 b?Q`iHv- pB bQ@+HH2/ ?KQ/B}+iBQMb?X
uQm rBHH b22 BM i?Bb Hi2bi p2`bBQM Q7 i?2 KQ/2H i?i i?2 pHm2b 7Q` i?2 M2rHv BMi`Q/m+2/ T`K2i2`b
`2 i?2 bK2 b i?2 ?`/@+Q/2/ pHm2b mb2/ 2`HB2`X b `2bmHi- i?2 /27mHi BMBiBH +QM/BiBQMb rBHH #2
2t+iHv i?2 bK2 b i?2v r2`2 #27Q`2X "mi MQr- r2 ?p2 i?2 7`22/QK iQ 2tTHQ`2 Qi?2` BMBiBH +QM/BiBQMb b
r2HHX 6Q` 2tKTH2- B7 r2 bBKmHi2 i?2a2+QM/P`/2`avbi2KAMBiS`KbKQ/2H b Bb- r2 ;2i i?2 7QHHQrBM;
bQHmiBQM 7Q` i?2 M;mH` TQbBiBQMb M/ p2HQ+BiB2b,
Re *?Ti2` RX .2b+`B#BM; "2?pBQ`

27. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
>Qr2p2`- B7 r2 KQ/B7v i?2T?BRnBMBiT`K2i2` iQ #2Ri i?2 bi`i Q7 Qm` bBKmHiBQM- r2 ;2i i?Bb
bQHmiBQM BMbi2/,
RXRX "bB+ 1[miBQMb Rd

28. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
1tTM/BM; QM i?Bb K2+?MB+H 2tKTH2
A7 vQm rQmH/ HBF2 iQ b22 i?Bb 2tKTH2 7m`i?2` /2p2HQT2/- vQm Kv rBb? iQ DmKT iQ i?2 b2i Q7 2tKTH2b
BMpQHpBM; `QiiBQMH bvbi2Kb 7QmM/ BM i?2 b2+iBQM QMaT22/ J2bm`2K2MiUT;2 ejVX
Pi?2`rBb2- vQm +M +QMiBMm2 iQ i?2 M2ti b2i Q7 2tKTH2b r?B+? BMpQHp2 TQTmHiBQM /vMKB+bX
GQiF@oQHi2`` avbi2Kb
aQ 7`- r2?p2 b22M i?2`KH- 2H2+i`B+H M/ K2+?MB+H 2tKTH2bX AM 2z2+i- i?2b2 ?p2 HH #22M 2M;BM22`BM;
2tKTH2bX >Qr2p2`- JQ/2HB+ Bb MQi HBKBi2/ bi`B+iHv iQ 2M;BM22`BM; /Bb+BTHBM2bX hQ `2BM7Q`+2 i?Bb TQBMi-
i?Bb b2+iBQM rBHH T`2b2Mi +QKKQM 2+QHQ;B+H bvbi2K /vMKB+b KQ/2H #b2/ QM i?2 `2HiBQMb?BT #2ir22M
T`2/iQ` M/ T`2v bT2+B2bX h?2 2[miBQMb r2 rBHH #2 mbBM; `2 i?2(GQiF)UT;2 j8dV@(oQHi2``)UT;2 j8dV
2[miBQMbX
*HbbB+ GQiF@oQHi2``
h?2+HbbB+ GQiF@oQHi2`` KQ/2H
3
BMpQHp2b irQ bT2+B2bX PM2 bT2+B2b Bb i?2 ?T`2v? bT2+B2bX AM i?Bb b2+iBQM-
i?2 TQTmHiBQM Q7 i?2 T`2v bT2+B2b rBHH #2 `2T`2b2Mi2/ #vxX h?2 Qi?2` bT2+B2b Bb i?2 ?T`2/iQ`? bT2+B2b
r?Qb2 TQTmHiBQM rBHH #2 `2T`2b2Mi2/ #vyX
h?2`2 `2 i?`22 BKTQ`iMi 2z2+ib BM GQiF@oQHi2`` bvbi2KX h?2 }`bi Bb `2T`Q/m+iBQM Q7 i?2 ?T`2v?
bT2+B2bX Ai Bb bbmK2/ i?i `2T`Q/m+iBQM Bb T`QTQ`iBQMH iQ i?2 TQTmHiBQMX A7 vQm `2 7KBHB` rBi?
+?2KB+H `2+iBQMb- i?Bb Bb +QM+2TimHHv i?2 bK2 b i?2Gr Q7 Jbb +iBQM
N
(:mH/#2`;)UT;2 j8dVX A7
3
?iiT,ff2MXrBFBT2/BXQ`;frBFBfGQiF@oQHi2``n2[miBQM
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?iiT,ff2MXrBFBT2/BXQ`;frBFBfGrnQ7nKbbn+iBQM
R3 *?Ti2` RX .2b+`B#BM; "2?pBQ`

29. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
vQm `2M?i 7KBHB` rBi? i?2 Gr Q7 Jbb +iBQM- Dmbi +QMbB/2` i?i i?2 KQ`2 TQi2MiBH Ki2b `2 T`2b2Mi
BM i?2 2MpB`QMK2Mi- i?2 KQ`2 HBF2Hv `2T`Q/m+iBQM Bb iQ Q++m`X q2 +M `2T`2b2Mi i?Bb Ki?2KiB+HHv b,
˙xr=αx
r?2`2xBb i?2 T`2v TQTmHiBQM-˙xrBb i?2 +?M;2 BM T`2v TQTmHiBQM/m2 iQ `2T`Q/m+iBQMM/αBb i?2
T`QTQ`iBQMHBiv +QMbiMi +Tim`BM; i?2 HBF2HB?QQ/ Q7 bm++2bb7mH `2T`Q/m+iBQMX
h?2 M2ti 2z2+i iQ +QMbB/2` Bb bi`piBQM Q7 i?2 T`2/iQ` bT2+B2bX A7 i?2`2 `2M?i 2MQm;? ?T`2v? `QmM/
iQ 2i- i?2 T`2/iQ` bT2+B2b rBHH /B2 QzX q?2M KQ/2HBM; bi`piBQM- Bi Bb BKTQ`iMi iQ +QMbB/2` i?2 2z2+i
Q7 +QKT2iBiBQMX q2 ;BM ?p2 T`QTQ`iBQMHBiv `2HiBQMb?BT- #mi i?Bb iBK2 Bi rQ`Fb BM `2p2`b2 #2+mb2-
mMHBF2 rBi? T`2v `2T`Q/m+iBQM- i?2 KQ`2 T`2/iQ`b `QmM/ i?2 KQ`2 HBF2Hv bi`piBQM BbX h?Bb Bb 2tT`2bb2/
Ki?2KiB+HHv BM Km+? i?2 bK2 rv b `2T`Q/m+iBQM,
˙ys=−γy
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bHB;?iHv /Bz2`2Mi bi`m+im`2 i?M `2T`Q/m+iBQM M/ bi`piBQM-BX2X-
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˙yp=δxy
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7QHHQrBM; irQ 2[miBQMb,
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˙y=˙yp+˙ys
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BMiQ,
˙x=x(α−βy)
˙y=y(δx−γ)
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_2HtUbi`i4tyV]S`2v TQTmHiBQM]c
_2HvUbi`i4vyV]S`2/iQ` TQTmHiBQM]c
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31. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
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p`B#H2 /m`BM; i?2 BMBiBH Bi2`iBQM Q7 i?2 MQM@HBM2` bQHp2`VX
q?2i?2` iQ bT2+B7v bi`iii`B#mi2 Q` MQi /2T2M/b QM ?Qr bi`B+iHv vQm rMi ;Bp2M BMBiBH +QM/BiBQM iQ
#2 2M7Q`+2/X EMQrBM; i?i Bb bQK2i?BM; i?i iF2b 2tT2`B2M+2 rQ`FBM; rBi? i?2 HM;m;2 M/ Bb #2vQM/
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rBi? #bB+ 2tTHMiBQM Q7 i?2 i`/2@QzbX
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7Q` i?2 GQiF@oQHi2`` bvbi2K +M #2 b22M BM i?Bb THQi,
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32. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
ai2/v aii2 AMBiBHBxiBQM
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˙x=0
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40. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
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43. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
JQ/B}+iBQMb
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_2HtUbi`i4RyVc
AM i?Bb +b2- r2 `2 +`2iBM; p`B#H2tQ7 ivT2_2HX "mi `i?2` i?M H2pBM; Bi ?b Bb?- r2 i?2M TTHv
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2tKTH2- r2 `2 QMHv ;QBM; QM2 H2p2H BMiQtiQ KF2 Qm` KQ/B}+iBQMX "mi b r2 rBHH b22 BM Qm` M2ti
2tKTH2- Bi Bb TQbbB#H2 iQ KF2 KQ/B}+iBQMb i `#Bi``v /2Ti?bX
h?2 Qi?2` +b2 r?2`2 r2 ?p2 b22M KQ/B}+iBQMb rb BM i?2 b2+iBQM QMpQB/BM; _2T2iBiBQMUT;2 k8VX
h?2`2 r2 br KQ/B}+iBQM mb2/ BM +QMDmM+iBQM rBi?2ti2M/b+Hmb2b-2X;X-
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44. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
AMBiBH 1[miBQMb
b r2 br BM KMv Q7 i?2 2tKTH2b BM i?Bb +?Ti2`- Bi Bb TQbbB#H2 iQ bT2+B7v 2[miBQMb rBi?BM KQ/2H iQ
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˙
⃗x(t)=
⃗
f(⃗x(t),⃗u(t),t)
⃗y(t)=⃗g(⃗x(t),⃗u(t),t)
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bii2b BM i?2 bvbi2KX PM2 }MH i?BM; iQ MQi2 #Qmi i?Bb bvbi2K Bb i?i M2Bi?2` 7mM+iBQM-
⃗
fMQ`⃗g- /2T2M/b
QM⃗yX
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i?2KX h?2 `2bQM i?i r2 +HH⃗xi?2 bii2 Q7 Qm` bvbi2K Bb i?i Bi i?2 QMHv BM7Q`KiBQM U7`QK rBi?BM
Qm` bvbi2KV M22/2/ iQ +QKTmi2
˙
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Q`/2` iQ ``Bp2 i bQHmiBQMVX
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˙
⃗x(t)iQ +QKTmi2⃗x(t)X AM Qi?2` rQ`/b,
⃗x(T)=
"
T
ti
˙
⃗x(t)/t+⃗x(ti)
h?Bb HH rQ`Fb b HQM; b i?2`2rb T`2pBQmb iBK2 bi2TX q?2M i?2`2 rbM?i T`2pBQmb iBK2 bi2T- i?2M
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Qi?2` rQ`/b- Qm` BMBiBH +QM/BiBQMbX
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⃗x(t0)=⃗x0
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AM i?2 b2+iBQM #Qp2 r2 /Q MQi BM+Hm/2 `2T`2b2MiiBQM Q7 i?2 GQiF@oQHi2`` 2[miBQMb BM GhA 7Q`KX h?Bb
Bb #2+mb2 i?2 GQiF@oQHi2`` 2[miBQMb- r?BH2 #2BM; iBK2@BMp`BMi- `2 MQi HBM2`X Ai Bb rQ`i? TQBMiBM;
Qmi i?i JQ/2HB+ /Q2b MQi /B`2+iHv 2M7Q`+2 2Bi?2` Q7 i?2b2 T`QT2`iB2b r?2M mbBM; i?2GhAKQ/2HX aQ Bi Bb
TQbbB#H2 iQ `2T`2b2Mi MQM@HBM2` Q` iBK2@p`BMi KQ/2Hb mbBM; i?Bb TT`Q+?X "mi Bi rQmH/ #2 +QM7mbBM;
bBM+2 i?2 i2`K GhA BKTHB2b i?i i?2 2[miBQMb `2 #Qi? HBM2` M/ iBK2@BMp`BMiX
lbBM; *QKTQM2Mib
AM HH Q7 i?2b2 2tKTH2b bQ 7`- r2?p2 mb2/ BM?2`BiM+2 UpB2ti2M/bV iQ `2mb2 i?2 2[miBQMb 7`QK i?2GhA
KQ/2HX AM ;2M2`H- i?2`2 Bb Km+? #2ii2`rv iQ `2mb2 i?2b2 2[miBQMb r?B+? Bb iQi`2i i?2K b
bm#@+QKTQM2MibX hQ b22 ?Qr i?Bb Bb /QM2- r2 rBHH `2+bi Qm` T`2pBQmb2H2+i`B+H 2tKTH2bUT;2 RyV BM
GhA 7Q`KX "mi i?Bb iBK2- r2?HH +`2i2 MK2/ BMbiM+2 Q7 i?2GhAKQ/2H,
KQ/2H_G*]aii2 bT+2 p2`bBQM Q7 M _G* +B`+mBi]
T`K2i2`_2Ho#4k9c
T`K2i2`_2HG4Rc
T`K2i2`_2H_4Ryyc
T`K2i2`_2H*4R2@jc
GhA `H+n+QKTUMt4k- Mm4R- Mv4k- ty4&y-y'-
4(@RfU_ *V-Rf*c@RfG-y)-
"4(ycRfG)-
*4(Rf_-yc@Rf_-R)-
.4(ycy)Vc
2[miBQM
`H+n+QKTXm4&o#'c
2M/_G*c
LQi2 i?i i?Bb iBK2 r2 /Q MQi mb22ti2M/bQ` BM?2`BiM+2 Q7 Mv FBM/X AMbi2/- r2 +imHHv /2+H`2
p`B#H2 +HH2/`H+n+QKTi?i Bb Q7 ivT2GhAX PM+2 r2 ?p2 }MBb?2/ +Qp2`BM; HH i?2 #bB+b Q7 ?Qr iQ
/2b+`B#2 /Bz2`2Mi FBM/b Q7 #2?pBQ` BM JQ/2HB+- r2?HH im`M Qm` ii2MiBQM iQ ?Qr iQ Q`;MBx2 HH i?2b2
2[miBQMb BMiQ `2mb#H2*QKTQM2MibUT;2 R3jVX "mi 7Q` MQr- i?Bb Bb Dmbi ?bM2F T22F? Q7 U#B;V i?BM;b
iQ +QK2X
q?i r2 b22 BM i?Bb_G*2tKTH2 Bb i?i r2 MQr ?p2 p`B#H2 +HH2/`H+n+QKTM/ i?Bb +QKTQM2Mi-
BM im`M- ?b HH i?2 T`K2i2`b M/ p`B#H2b Q7 i?2GhAKQ/2H BMbB/2 BiX aQ- 7Q` 2tKTH2- r2 b22 i?i Qm`
2[miBQM iQ bT2+B7v i?2 BMTmi-m- Bb r`Bii2M b,
`H+n+QKTXm4&o#'c
LQi2 i?i i?Bb 2[miBQM K2Mb i?i r2 `2 T`QpB/BM; M 2[miBQM 7Q` i?2 p`B#H2mi?i BbBMbB/2i?2
p`B#H2`H+n+QKTX b r2 rBHH b22 Hi2`- r2 +M mb2 ?B2``+?v iQ KM;2 +QMbB/2`#H2 KQmMi Q7
+QKTH2tBiv i?i `Bb2b 7`QK +QKTH2t bvbi2K /2b+`BTiBQMbX h?2 mb2 Q7 i?2XQT2`iBQM ?2`2 Bb ?Qr r2
+M `272`2M+2 p`B#H2b i?i `2 Q`;MBx2/ BM i?Bb ?B2``+?B+H KMM2`X ;BM- i?Bb rBHH #2 /Bb+mbb2/
i?Q`Qm;?Hv r?2M r2 BMi`Q/m+2*QKTQM2MibUT;2 R3jVX
[email protected] >2i h`Mb72`
Pm` T`2pBQmb /Bb+mbbBQM QMaii2 aT+2UT;2 3dV BMi`Q/m+2/ Ki`B+2b M/ p2+iQ`bX h?2 7Q+mb rb
T`BK`BHv QM Ki?2KiB+H bT2+ib Q7 ``vbX AM i?Bb b2+iBQM- r2 rBHH +QMbB/2` ?Qr ``vb +M #2 mb2/ iQ
`2T`2b2Mi bQK2i?BM; #Bi KQ`2 T?vbB+H- i?2 QM2@/BK2MbBQMH bTiBH /Bbi`B#miBQM Q7 p`B#H2bX q2?HH HQQF
RXjX o2+iQ`b M/ ``vb NR

102. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
i b2p2`H 72im`2b BM JQ/2HB+ i?i `2 `2Hi2/ iQ ``vb M/ ?Qr i?2v HHQr mb iQ +QKT+iHv 2tT`2bb
#2?pBQ` BMpQHpBM; ``vbX
Pm` T`Q#H2Kb rBHH +2Mi2` `QmM/ i?2 bBKTH2 ?2i i`Mb72` T`Q#H2KX *QMbB/2` QM2@/BK2MbBQMH `Q/
HBF2 i?2 QM2 b?QrM #2HQr,
.2`BpBM; 1[miBQMb
"27Q`2 ;2iiBM; BMiQ i?2 Ki?- i?2`2 Bb M BKTQ`iMi TQBMi rQ`i? KFBM;X JQ/2HB+ Bb HM;m;2 7Q`
`2T`2b2MiBM;HmKT2/bvbi2KbX q?i i?Bb K2Mb Bb i?i i?2 #2?pBQ` Kmbi #2 2tT`2bb2/ BM i2`Kb Q7
Q`/BM`v /Bz2`2MiBH 2[miBQMb UP.1bV Q` BM bQK2 +b2b /Bz2`2MiBH@H;2#`B+ 2[miBQMb U.1bVX "mi
JQ/2HB+ /Q2b MQi BM+Hm/2 Mv K2Mb 7Q` /2b+`B#BM; T`iBH /Bz2`2MiBH 2[miBQMb UBX2X-2[miBQMb i?i
BMpQHp2 i?2 ;`/B2Mi Q7 p`B#H2b BM bTiBH /B`2+iBQMbVX b bm+?- BM i?Bb b2+iBQM r2 rBHH /2`Bp2 i?2 Q`/BM`v
/Bz2`2MiBH 2[miBQMb i?i `2bmHi r?2M r2 /Bb+`2iBx2 `Q/ BMiQ /Bb+`2i2 TB2+2b M/ i?2M KQ/2H i?2 #2?pBQ`
Q7 2+? Q7 i?2b2 /Bb+`2i2 UHmKT2/V TB2+2bX
qBi? i?i Qmi Q7 i?2 rv- H2i mb +QMbB/2` i?2 ?2i #HM+2 7Q` 2+? /Bb+`2i2 b2+iBQM Q7 i?Bb `Q/X 6B`bi- r2
?p2 i?2 i?2`KH +T+BiM+2 Q7 i?2i
th
b2+iBQMX h?Bb +M 2tT`2bb2/ b,
miCTi
r?2`2mBb i?2 Kbb Q7 i?2i
th
b2+iBQM-CBb i?2 +T+BiM+2 U Ki2`BH T`QT2`ivV M/TiBb i?2 i2KT2`im`2
Q7 i?2i
th
b2+iBQMX q2 +M 7m`i?2` /2b+`B#2 i?2 Kbb b,
mi=ρVi
r?2`2ρBb i?2 Ki2`BH /2MbBiv M/ViBb i?2 pQHmK2 Q7 i?2i
th
b2+iBQMX 6BMHHv- r2 +M 2tT`2bb i?2 pQHmK2
Q7 i?2i
th
b2+iBQM b,
Vi=AcLi
r?2`2AcBb i?2 +`Qbb@b2+iBQMH `2 Q7 i?2i
th
b2+iBQM- r?B+? Bb bbmK2/ iQ #2 mMB7Q`K- M/LiBb i?2
H2M;i? Q7 i?2i
th
b2+iBQMX 6Q` i?Bb 2tKTH2- r2 rBHH bbmK2 i?2 `Q/ Bb +QKTQb2/ Q7 2[mH bBx2 TB2+2bX AM
i?Bb +b2- r2 +M /2}M2 i?2 b2;K2Mi H2M;i?-Li- iQ #2,
Li=
L
n
q2 rBHH HbQ bbmK2 i?i i?2 +`Qbb@b2+iBQMH `2 Bb mMB7Q`K HQM; i?2 H2M;i? Q7 i?2 `Q/X b bm+?- i?2
Kbb Q7 2+? b2;K2Mi +M #2 ;Bp2M b,
m=ρAcLi
AM i?Bb +b2- i?2 i?2`KH +T+BiM+2 Q7 2+? b2+iBQM rQmH/ #2,
ρAcLiCTi
h?Bb- BM im`M- K2Mb i?i i?2 M2i ?2i ;BM2/ BM i?i b2+iBQM i Mv iBK2 rBHH #2,
ρAcLiC
/Ti
/t
Nk *?Ti2` RX .2b+`B#BM; "2?pBQ`

103. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
r?2`2 r2 bbmK2 i?iAc-LiM/C/QM?i +?M;2 rBi? `2bT2+i iQ iBK2X
h?i +Qp2`b i?2 i?2`KH +T+BiM+2X AM //BiBQM- r2 rBHH +QMbB/2` irQ /Bz2`2Mi 7Q`Kb Q7 ?2i i`Mb@
72`X h?2 }`bi 7Q`K Q7 ?2i i`Mb72` r2 rBHH +QMbB/2` Bb +QMp2+iBQM 7`QK 2+? b2+iBQM iQ bQK2 K#B2Mi
i2KT2`im`2-TambX q2 +M 2tT`2bb i?2 KQmMi Q7 ?2i HQbi 7`QK 2+? b2+iBQM b,
qh=−hAsi(Ti−Tamb)
r?2`2hBb i?2 +QMp2+iBQM +Q2{+B2Mi M/AsiBb i?2 bm`7+2 `2 Q7 i?2i
th
b2+iBQMX h?2 Qi?2` 7Q`K Q7
?2i i`Mb72` Bb +QM/m+iBQM iQ M2B;?#Q`BM; b2+iBQMbX >2`2 i?2`2 rBHH #2 irQ +QMi`B#miBQMb- QM2 HQbi iQ i?2
i−1
th
b2+iBQM- B7 Bi 2tBbib- M/ i?2 Qi?2` HQbi iQ i?2i+1
th
b2+iBQM- B7 Bi 2tBbibX h?2b2 +M #2 `2T`2b2Mi2/-
`2bT2+iBp2Hv- b,
qki→i−1=−kAc
Ti−Ti−1
Li
qki→i+1=−kAc
Ti−Ti+1
Li
lbBM; i?2b2 `2HiBQMb- r2 FMQr i?i i?2 ?2i #HM+2 7Q` i?2 }`bi 2H2K2Mi rQmH/ #2,
ρAcLiC
/T1
/t
=−hAsi(T1−Tamb)−kAc
T1−T2
Li
aBKBH`Hv- i?2 ?2i #HM+2 7Q` i?2 Hbi 2H2K2Mi rQmH/ #2,
ρAcLiC
/Tn
/t
=−hAsn(Tn−Tamb)−kAc
Tn−Tn−1
Li
6BMHHv- i?2 ?2i #HM+2 7Q` HH Qi?2` 2H2K2Mib rQmH/ #2,
ρAcLiC
/Ti
/t
=−hAsi(Ti−Tamb)−kAc
Ti−Ti−1
Li
−kAc
Ti−Ti+1
Li
AKTH2K2MiiBQM
q2 bi`i #v /2}MBM; ivT2b 7Q` i?2 p`BQmb T?vbB+H [mMiBiB2bX h?Bb rBHH ;Bp2 mb i?2 T`QT2` mMBib M/-
/2T2M/BM; QM i?2 iQQH- HHQrb mb iQ /Q mMBi +?2+FBM; QM Qm` 2[miBQMbX Pm` ivT2 /2}MBiBQMb `2 b 7QHHQrb,
ivT2h2KT2`im`24_2HUmMBi4]E]-KBM4yVc
ivT2*QMp2+iBQM*Q277B+B2Mi4_2HUmMBi4]qfE]-KBM4yVc
ivT2*QM/m+iBQM*Q277B+B2Mi4_2HUmMBi4]qXK@RXE@R]-KBM4yVc
ivT2Jbb4_2HUmMBi4]F;]-KBM4yVc
ivT2aT2+B7B+>2i4_2HUmMBi4]CfUEXF;V]-KBM4yVc
ivT2.2MbBiv4_2HUmMBi4]F;fKj]-KBM4yVc
ivT2`24_2HUmMBi4]Kk]Vc
ivT2oQHmK24_2HUmMBi4]Kj]Vc
ivT2G2M;i?4_2HUmMBi4]K]-KBM4yVc
ivT2_/Bmb4_2HUmMBi4]K]-KBM4yVc
q2 rBHH HbQ /2}M2 b2p2`H T`K2i2`b iQ /2b+`B#2 i?2 `Q/ r2 `2 bBKmHiBM;,
T`K2i2`AMi2;2`M4Ryc
T`K2i2`G2M;i? G4RXyc
T`K2i2`_/Bmb _4yXRc
T`K2i2`.2MbBiv `?Q4kXyc
T`K2i2`*QMp2+iBQM*Q277B+B2Mi ?4kXyc
T`K2i2`*QM/m+iBQM*Q277B+B2Mi F4Ryc
T`K2i2`aT2+B7B+>2i *4RyXyc
T`K2i2`h2KT2`im`2 hK#4jyy]K#B2Mi i2KT2`im`2]c
:Bp2M i?2b2 T`K2i2`b- r2 +M +QKTmi2 i?2 `2b M/ pQHmK2 7Q` 2+? b2+iBQM BM i2`Kb Q7 i?2 T`K2i2`b
r2 ?p2 H`2/v /2}M2/ mbBM; i?2 7QHHQrBM; /2+H`iBQMb,
RXjX o2+iQ`b M/ ``vb Nj

104. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
T`K2i2``2 n+4TB _k- nb4k TB _ Gc
T`K2i2`oQHmK2 o4n+ GfMc
6BMHHv- i?2 QMHv ``v BM i?Bb T`Q#H2K Bb i?2 i2KT2`im`2 Q7 2+? b2+iBQM UbBM+2 i?Bb Bb i?2 QMHv [mMiBiv
i?i +imHHv p`B2b HQM; i?2 H2M;i? Q7 i?2 `Q/V,
h2KT2`im`2 h(M)c
h?Bb +QM+Hm/2b HH i?2 /2+H`iBQMb r2 M22/ iQ KF2X LQr H2i?b +QMbB/2` i?2 p`BQmb 2[miBQMb `2[mB`2/X
6B`bi- r2 M22/ iQ bT2+B7v i?2 BMBiBH +QM/BiBQMb 7Q` i?2 `Q/X q2 rBHH bbmK2 i?iT1(0) = 200-Tn(0) = 300
M/ i?2 BMBiBH i2KT2`im`2b Q7 HH Qi?2` b2+iBQMb +M #2 HBM2`Hv BMi2`TQHi2/ #2ir22M i?2b2 irQ 2M/
+QM/BiBQMbX h?Bb Bb +Tim`2/ #v i?2 7QHHQrBM; 2[miBQM,
BMBiBH2[miBQM
h4HBMbT+2Ukyy-jyy-MVc
r?2`2 i?2HBMbT+2QT2`iQ` Bb mb2/ iQ +`2i2 M ``v Q7MpHm2b i?i p`v HBM2`Hv #2ir22MkyyM/
jyyX _2+HH 7`QK Qm`aii2 aT+2UT;2 3dV 2tKTH2b i?i r2 +M BM+Hm/2 2[miBQMb r?2`2 i?2 H27i ?M/
bB/2 M/ `B;?i ?M/ bB/2 2tT`2bbBQMb `2 p2+iQ`bX h?Bb Bb MQi?2` 2tKTH2 Q7 bm+? M 2[miBQMX
6BMHHv- r2 +QK2 iQ i?2 2[miBQMb i?i /2b+`B#2 ?Qr i?2 i2KT2`im`2 BM 2+? b2+iBQM +?M;2b Qp2` iBK2,
2[miBQM
`?Q o * /2`Uh(R)V4@? nb Uh(R)@hK#V@F n+ Uh(R)@h(k)VfUGfMVc
7Q`BBMk,UM@RVHQQT
`?Q o * /2`Uh(B)V4@? nb Uh(B)@hK#V@F n+ Uh(B)@h(B@R)VfUGfMV@F n+ Uh(B)@h(BYR)VfUGfMVc
2M/7Q`c
`?Q o * /2`Uh(2M/)V4@? nb Uh(2M/)@hK#V@F n+ Uh(2M/)@h(2M/@R)VfUGfMVc
h?2 }`bi 2[miBQM +Q``2bTQM/b iQ i?2 ?2i #HM+2 7Q` b2+iBQM1- i?2 Hbi 2[miBQM +Q``2bTQM/b iQ i?2
?2i #HM+2 7Q` b2+iBQMnM/ i?2 KB//H2 2[miBQM +Qp2`b HH Qi?2` b2+iBQMbX LQi2 i?2 mb2 Q72M/b
bm#b+`BTiX q?2M M 2tT`2bbBQM Bb mb2/ iQ 2pHmi2 bm#b+`BTi 7Q` ;Bp2M /BK2MbBQM-2M/`2T`2b2Mib i?2
bBx2 Q7 i?i /BK2MbBQMX AM Qm` +b2- r2 mb22M/iQ `2T`2b2Mi i?2 Hbi b2+iBQMX P7 +Qm`b2- r2 +QmH/ mb2MBM
i?Bb +b2- #mi BM ;2M2`H-2M/+M #2 p2`v mb27mH r?2M i?2 bBx2 Q7 /BK2MbBQM Bb MQi H`2/v bbQ+Bi2/
rBi? p`B#H2X
HbQ MQi2 i?2 mb2 Q7 7Q`HQQT BM i?Bb KQ/2HX 7Q`HQQT HHQrb i?2 HQQT BM/2t p`B#H2 iQ HQQT Qp2`
`M;2 Q7 pHm2bX AM Qm` +b2- i?2 HQQT BM/2t p`B#H2 BbBM/ i?2 `M;2 Q7 pHm2b Bb2i?`Qm;?n−1X h?2
;2M2`H bvMit 7Q` 7Q`HQQT Bb,
7Q`Ip`=BMI`M;2=HQQT
ff bii2K2Mib
2M/7Q`c
r?2`2I`M;2=Bb p2+iQ` Q7 pHm2bX +QMp2MB2Mi rv iQ ;2M2`i2 b2[m2M+2 Q7 pHm2b Bb iQ mb2 i?2
`M;2 QT2`iQ`-,X h?2 pHm2 #27Q`2 i?2 `M;2 QT2`iQ` Bb i?2 BMBiBH pHm2 BM i?2 b2[m2M+2 M/ i?2 pHm2
7i2` i?2 `M;2 QT2`iQ` Bb i?2 }MH pHm2 BM i?2 b2[m2M+2X aQ- 7Q` 2tKTH2- i?2 2tT`2bbBQM8,RyrQmH/
;2M2`i2 p2+iQ` rBi? i?2 pHm2b8-e-d-3-NM/RyX LQi2 i?i i?BbBM+Hm/2bi?2 pHm2b mb2/ iQ bT2+B7v
i?2 `M;2X
q?2M 7Q`HQQT Bb mb2/ BM M 2[miBQM b2+iBQM- 2+? Bi2`iBQM Q7 i?2 7Q` HQQT ;2M2`i2b M2r 2[miBQM
7Q` 2+? 2[miBQM BMbB/2 i?27Q`HQQTX aQ BM Qm` +b2- r2 rBHH ;2M2`i2n−22[miBQMb +Q``2bTQM/BM; iQ
pHm2b Q7B#2ir22MkM/M@RX
SmiiBM; HH i?Bb iQ;2i?2`- i?2 +QKTH2i2 KQ/2H rQmH/ #2,
KQ/2H_Q/n6Q`GQQT]JQ/2HBM; ?2i +QM/m+iBQM BM `Q/ mbBM; 7Q` HQQT]
ivT2h2KT2`im`24_2HUmMBi4]E]-KBM4yVc
ivT2*QMp2+iBQM*Q277B+B2Mi4_2HUmMBi4]qfE]-KBM4yVc
ivT2*QM/m+iBQM*Q277B+B2Mi4_2HUmMBi4]qXK@RXE@R]-KBM4yVc
ivT2Jbb4_2HUmMBi4]F;]-KBM4yVc
ivT2aT2+B7B+>2i4_2HUmMBi4]CfUEXF;V]-KBM4yVc
N9 *?Ti2` RX .2b+`B#BM; "2?pBQ`

105. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
ivT2.2MbBiv4_2HUmMBi4]F;fKj]-KBM4yVc
ivT2`24_2HUmMBi4]Kk]Vc
ivT2oQHmK24_2HUmMBi4]Kj]Vc
ivT2G2M;i?4_2HUmMBi4]K]-KBM4yVc
ivT2_/Bmb4_2HUmMBi4]K]-KBM4yVc
+QMbiMi_2HTB4jXR9R8Nc
T`K2i2`AMi2;2`M4Ryc
T`K2i2`G2M;i? G4RXyc
T`K2i2`_/Bmb _4yXRc
T`K2i2`.2MbBiv `?Q4kXyc
T`K2i2`*QMp2+iBQM*Q277B+B2Mi ?4kXyc
T`K2i2`*QM/m+iBQM*Q277B+B2Mi F4Ryc
T`K2i2`aT2+B7B+>2i *4RyXyc
T`K2i2`h2KT2`im`2 hK#4jyy]K#B2Mi i2KT2`im`2]c
T`K2i2``2 n+4TB _k- nb4k TB _ Gc
T`K2i2`oQHmK2 o4n+ GfMc
h2KT2`im`2 h(M)c
BMBiBH2[miBQM
h4HBMbT+2Ukyy-jyy-MVc
2[miBQM
`?Q o * /2`Uh(R)V4@? nb Uh(R)@hK#V@F n+ Uh(R)@h(k)VfUGfMVc
7Q`BBMk,UM@RVHQQT
`?Q o * /2`Uh(B)V4@? nb Uh(B)@hK#V@F n+ Uh(B)@h(B@R)VfUGfMV@F n+ Uh(B)@h(BYR)VfUGfMVc
2M/7Q`c
`?Q o * /2`Uh(2M/)V4@? nb Uh(2M/)@hK#V@F n+ Uh(2M/)@h(2M/@R)VfUGfMVc
2M/_Q/n6Q`GQQTc
LQi2,LQi2 i?i r2?p2 BM+Hm/2/TBb HBi2`H +QMbiMi BM i?Bb KQ/2HX Gi2` BM i?2 #QQF- r2?HH /Bb+mbb
?Qr iQ T`QT2`Hv BKTQ`i +QKKQM*QMbiMibUT;2 RejVX
aBKmHiBM; i?Bb KQ/2H vB2H/b i?2 7QHHQrBM; bQHmiBQM 7Q` 2+? Q7 i?2 MQ/H i2KT2`im`2b,
RXjX o2+iQ`b M/ ``vb N8

106. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
LQi2 ?Qr i?2 i2KT2`im`2b `2 BMBiBHHv /Bbi`B#mi2/ HBM2`Hv Ub r2 bT2+B}2/ BM Qm`BMBiBH 2[miBQM
b2+iBQMVX
Hi2`MiBp2b
Ai im`Mb Qmi i?i i?2`2 `2 b2p2`H rvb r2 +M ;2M2`i2 i?2 2[miBQMb r2 M22/X 1+? ?b Bib QrM
/pMi;2b M/ /Bb/pMi;2b /2T2M/BM; QM i?2 +QMi2tiX q2?HH T`2b2Mi i?2K ?2`2 Dmbi iQ /2KQMbi`i2
i?2 TQbbB#BHBiB2bX h?2 +?QB+2 Q7 r?B+? QM2 i?2v 722H H2/b iQ i?2 KQbi mM/2`biM/#H2 2[miBQMb Bb mT iQ
i?2 KQ/2H /2p2HQT2`X
PM2 ``v 72im`2 r2 +M mb2 iQ KF2 i?2b2 2[miBQMb bHB;?iHv bBKTH2` Bb +HH2/ M ``v +QKT`2?2MbBQMX
M ``v +QKT`2?2MbBQM ~BTb i?27Q`HQQT `QmM/ bQ i?i r2 iF2 bBM;H2 2[miBQM M/ // bQK2
BM7Q`KiBQM i i?2 2M/ BM/B+iBM; i?i i?2 2[miBQM b?QmH/ #2 2pHmi2/ 7Q` /Bz2`2Mi pHm2b Q7 i?2 HQQT
BM/2t p`B#H2X AM Qm` +b2- r2 +M `2T`2b2Mi Qm` 2[miBQM b2+iBQM mbBM; ``v +QKT`2?2MbBQMb b 7QHHQrb,
2[miBQM
`?Q o * /2`Uh(R)V4@? nb Uh(R)@hK#V@F n+ Uh(R)@h(k)VfUGfMVc
`?Q o * /2`Uh(k,M@R)V4&@? nb Uh(B)@hK#V@F n+ Uh(B)@h(B@R)VfUGfMV@F n+ Uh(B)@h(BYR)VfUGf
!→MV7Q`BBMk,UM@RV'c
`?Q o * /2`Uh(2M/)V4@? nb Uh(2M/)@hK#V@F n+ Uh(2M/)@h(2M/@R)VfUGfMVc
q2 +QmH/ HbQ +QK#BM2 i?2 ``v +QKT`2?2MbBQM rBi? bQK2B72tT`2bbBQMb iQ MmHHB7v +QMi`B#miBQMb iQ
i?2 ?2i #HM+2 i?i /QM?i M2+2bb`BHv TTHvX AM i?i +b2- r2 +M bBKTHB7v i?22[miBQMb2+iBQM iQ i?2
TQBMi r?2`2 Bi +QMiBMb QM2 U/KBii2/Hv KmHiB@HBM2V 2[miBQM,
2[miBQM
`?Q o * /2`UhV4&@? nb Uh(B)@hK#V
@UB7B44Ri?2My2Hb2F n+fUGfMV Uh(B)@h(B@R)VV
@UB7B44Mi?2My2Hb2F n+fUGfMV Uh(B)@h(BYR)VV7Q`BBMR,M'c
Ne *?Ti2` RX .2b+`B#BM; "2?pBQ`

107. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
_2+HH- 7`QK b2p2`H T`2pBQmb 2tKTH2b- i?i JQ/2HB+ bmTTQ`ib p2+iQ` 2[miBQMbX AM i?2b2 +b2b- r?2M
i?2 H27i ?M/ M/ `B;?i ?M/ bB/2 `2 p2+iQ`b Q7 i?2 bK2 bBx2- r2 +M mb2 bBM;H2 Up2+iQ`V 2[miBQM iQ
`2T`2b2Mi KMv b+H` 2[miBQMbX q2 +M mb2 i?Bb 72im`2 iQ bBKTHB7v Qm` 2[miBQMb b 7QHHQrb,
2[miBQM
`?Q o * /2`Uh(R)V4@? nb Uh(R)@hK#V@F n+ Uh(R)@h(k)VfUGfMVc
`?Q o * /2`Uh(k,M@R)V4@? nb Uh(B)@hK#V@F n+ Uh(k,M@R)@h(R,M@k)VfUGfMV@F n+ Uh(k,M@R)@
!→h(j,M)VfUGfMVc
`?Q o * /2`Uh(2M/)V4@? nb Uh(2M/)@hK#V@F n+ Uh(2M/)@h(2M/@R)VfUGfMVc
LQi2 i?i r?2M p2+iQ` p`B#H2 HBF2h?b `M;2 Q7 bm#b+`BTib TTHB2/ iQ Bi- i?2 `2bmHi Bb p2+iQ`
+QMiBMBM; i?2 +QKTQM2Mib BM/B+i2/ #v i?2 pHm2b BM i?2 bm#b+`BTiX 6Q` 2tKTH2- i?2 2tT`2bbBQMh(k,9)
Bb 2[mBpH2Mi iQ&h(k)- h(j)- h(9)'X h?2 bm#b+`BTi 2tT`2bbBQM /Q2bM?i M22/ iQ #2 `M;2X 6Q` 2tKTH2-
h(&k-8-N')Bb 2[mBpH2Mi iQ&h(k)- h(8)- h(N)'X
6BMHHv- H2i mb +QMbB/2` QM2 Hbi rv Q7 `27+iQ`BM; i?2b2 2[miBQMbX AK;BM2 r2 BMi`Q/m+2/ i?`22 //BiBQMH
p2+iQ` p`B#H2b,
>2i Z+QMp(M)c
>2i ZH27i(M)c
>2i Z`B;?i(M)c
h?2M r2 +M r`Bi2 i?2b2 irQ 2[miBQMb U;BM mbBM; p2+iQ` 2[miBQMbV iQ /2}M2 i?2 ?2i HQbi iQ i?2
K#B2Mi- T`2pBQmb b2+iBQM M/ M2ti b2+iBQM BM i?2 `Q/,
Z+QMp4&@? nb Uh(B)@hK#V7Q`BBMR,M'c
ZH27i4&UB7B44Ri?2My2Hb2@F n+ Uh(B)@h(B@R)VfUGfMVV7Q`BBMR,M'c
Z`B;?i4&UB7B44Mi?2My2Hb2@F n+ Uh(B)@h(BYR)VfUGfMVV7Q`BBMR,M'c
h?Bb HHQrb mb iQ 2tT`2bb i?2 ?2i #HM+2 7Q` 2+? b2+iBQM mbBM; p2+iQ` 2[miBQM i?i /Q2bM?i BM+Hm/2
Mv bm#b+`BTib,
`?Q o * /2`UhV4Z+QMpYZH27iYZ`B;?ic
*QM+HmbBQM
AM i?Bb b2+iBQM- r2?p2 b22M p`BQmb rvb i?i r2 +M mb2 p2+iQ` p`B#H2b M/ p2+iQ` 2[miBQMb iQ `2T`2b2Mi
QM2@/BK2MbBQMH ?2i i`Mb72`X P7 +Qm`b2- i?Bb p2+iQ` `2Hi2/ 7mM+iBQMHBiv +M #2 mb2/ 7Q` rB/2 `M;2
Q7 /Bz2`2Mi T`Q#H2K ivT2bX h?2 ;QH Q7 i?Bb b2+iBQM rb iQ BMi`Q/m+2 b2p2`H 72im`2b iQ /2KQMbi`i2 i?2
p`BQmb QTiBQMb i?i `2 pBH#H2 iQ i?2 /2p2HQT2` r?2M rQ`FBM; rBi? p2+iQ`bX
*?2KB+H avbi2K
AM i?Bb b2+iBQM- r2?HH +QMbB/2` 72r /Bz2`2Mi rvb iQ /2b+`B#2 i?2 #2?pBQ` Q7 +?2KB+H bvbi2KX q2?HH
bi`i #v #mBH/BM; KQ/2H rBi?Qmi mbBM; i?2 ``v 7mM+iBQMHBivX h?2M- r2?HH BKTH2K2Mi i?2 bK2 #2?pBQ`
mbBM; p2+iQ`bX 6BMHHv- r2?HH BKTH2K2Mi i?2 bK2 KQ/2H ;BM mbBM; 2MmK2`iBQMbX
AM HH Q7 Qm` 2tKTH2b- r2?HH #2 #mBH/BM; KQ/2H 7Q`i?2 7QHHQrBM; bvbi2K Q7 `2+iBQMb
Rj
,
A+B→
1
X
A+B←
2
X
X+B→
3
T+S
Ai b?QmH/ #2 MQi2/ i?iXBb bBKTHv M BMi2`K2/Bi2 `2bmHi Q7 i?Bb `2+iBQMX h?2 Qp2`HH `2+iBQM +M #2
2tT`2bb2/ b,
A+2B→T+S
Rj
?iiT,ffHB#``vXrQH7`KX+QKfBM7Q+2Mi2`fh2+?LQi2bfjNyf
RXjX o2+iQ`b M/ ``vb Nd

108. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
lbBM; i?2 Hr Q7 Kbb +iBQM r2 +M i`Mb7Q`K i?2b2 +?2KB+H 2[miBQMb BMiQ i?2 7QHHQrBM; Ki?2KiB+H
QM2b,
/[A]
/t
=−k1[A][B]+k2[X]
/[B]
/t
=−k1[A][B]+k2[X]−k3[B][X]
/[X]
/t
=k1[A][B]−k2[X]−k3[B][X]
r?2`2k1-k2M/k3`2 i?2 `2+iBQM +Q2{+B2Mib 7Q` i?2 }`bi- b2+QM/ M/ i?B`/ `2+iBQMb- `2bT2+iBp2HvX
h?2b2 2[miBQMb `2 /2`Bp2/ #v +QMbB/2`BM; i?2 +?M;2 BM 2+? bT2+B2b /m2 iQ 2+? `2+iBQM BMpQHpBM;
i?i bT2+B2bX aQ- 7Q` 2tKTH2- bBM+2 i?2 }`bi `2+iBQMA+B→Xi`Mb7Q`Kb KQH2+mH2b Q7AM/BBMiQ
KQH2+mH2b Q7X- r2 b22 i?2 i2`K−k1[A][B]BM i?2 #HM+2 2[miBQM 7Q`A- r?B+? `2T`2b2Mib i?2 `2/m+iBQM
BM i?2 KQmMi Q7Ab `2bmHi Q7 i?i `2+iBQMX 1+? i2`K BM i?2b2 #HM+2 2[miBQMb Bb /2`Bp2/ BM
bBKBH` 7b?BQMX
qBi?Qmi ``vb
G2i mb bi`i rBi? M TT`Q+? i?i /Q2bM?i miBHBx2 ``vb i HHX AM i?Bb +b2- r2 bBKTHv `2T`2b2Mi i?2
+QM+2Mi`iBQMb[A]-[B]M/[X]#v i?2 p`B#H2b+-+"M/+sb 7QHHQrb,
KQ/2H_2+iBQMbnLQ``vb]JQ/2HBM; +?2KB+H `2+iBQM rBi?Qmi ``vb]
_2H+c
_2H+"c
_2H+sc
T`K2i2`_2HFR4yXRc
T`K2i2`_2HFk4yXRc
T`K2i2`_2HFj4Ryc
BMBiBH2[miBQM
+4Rc
+"4Rc
+s4yc
2[miBQM
/2`U+V4@FR + +"YFk +sc
/2`U+"V4@FR + +"YFk +s@Fj +" +sc
/2`U+sV4FR + +"@Fk +s@Fj +" +sc
2M/_2+iBQMbnLQ``vbc
qBi? i?Bb TT`Q+?- r2 +`2i2 M 2[miBQM 7Q` i?2 #HM+2 Q7 2+? bT2+B2bX A7 r2 bBKmHi2 i?Bb KQ/2H- r2
;2i i?2 7QHHQrBM; `2bmHib,
N3 *?Ti2` RX .2b+`B#BM; "2?pBQ`

109. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
lbBM; ``vb
MQi?2` rv iQ TT`Q+? KQ/2HBM; Q7 i?2 +?2KB+H bvbi2K Bb iQ mb2 p2+iQ`bX qBi? i?Bb TT`Q+?- r2
bbQ+Bi2/ i?2 bT2+B2bA-BM/XrBi? i?2 BM/B+2b1-2M/3- `2bT2+iBp2HvX h?2 +QM+2Mi`iBQMb `2
KTT2/ iQ i?2 p2+iQ` p`B#H2*X q2 +M HbQ +bi i?2 `2+iBQM +Q2{+B2Mib BMiQ p2+iQ` Q7 `2+iBQM
+Q2{+B2Mib-FX
qBi? i?Bb i`Mb7Q`KiBQM- HH i?2 2[miBQMb `2 i?2M i`Mb7Q`K2/ BMiQ p2+iQ` 2[miBQMb,
KQ/2H_2+iBQMbn``v]JQ/2HBM; +?2KB+H `2+iBQM rBi? ``vb]
_2H*(j)c
T`K2i2`_2HF(j)4&yXR-yXR-Ry'c
BMBiBH2[miBQM
*4&R-R-y'c
2[miBQM
/2`U*V4&@F(R) *(R) *(k)YF(k) *(j)-
@F(R) *(R) *(k)YF(k) *(j)@F(j) *(k) *(j)-
F(R) *(R) *(k)@F(k) *(j)@F(j) *(k) *(j)'c
2M/_2+iBQMbn``vc
h?2 `2+iBQM 2[miBQMb `2 MQM@HBM2`- bQ i?2v +MMQi #2 i`Mb7Q`K2/ BMiQ +QKTH2i2Hv HBM2` 7Q`KX "mi
r2 +QmH/ bBKTHB7v i?2K 7m`i?2` #v mbBM; Ki`Bt@p2+iQ` T`Q/m+iX AM Qi?2` rQ`/b- i?2 2[miBQMb,
/[A]
/t
=−k1[A][B]+k2[X]
/[B]
/t
=−k1[A][B]+k2[X]−k3[B][X]
/[X]
/t
=k1[A][B]−k2[X]−k3[B][X]
RXjX o2+iQ`b M/ ``vb NN

110. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
+M #2 i`Mb7Q`K2/ BMiQ i?2 7QHHQrBM; 7Q`K,
/
/t
⎧
⎨
⎩
[A]
[B]
[X]
⎫
⎬
⎭
=
⎡
⎣
−k1[B] 0k2
−k1[B]−k3[X]k2
k1[B]−k3[X]−k2
⎤
⎦
⎧
⎨
⎩
[A]
[B]
[X]
⎫
⎬
⎭
r?B+? +M i?2M #2 `2T`2b2Mi2/ BM JQ/2HB+ b,
/2`U*V4(@F(R) *(k)-y- F(k)c
@F(R) *(k)-@F(j) *(j)- F(k)c
F(R) *(k)-@F(j) *(j)-@F(k)) *c
h?2 /`r#+F Q7 i?Bb TT`Q+? Bb i?i r2 ?p2 iQ +QMbiMiHv F22T i`+F Q7 r?B+? BM/2t U2X;X-R-k- Q`jV
+Q``2bTQM/b iQ r?B+? bT2+B2b U2X;X-A-B- Q`XVX
lbBM; 1MmK2`iBQMb
hQ //`2bb i?Bb Bbbm2 Q7 ?pBM; iQ KT #+F M/ 7Q`i? 7`QK MmK#2`b iQ MK2b- Qm` i?B`/ TT`Q+? rBHH
miBHBx2 i?22MmK2`iBQMivT2 BM JQ/2HB+X M 2MmK2`iBQM HHQrb mb iQ /2}M2 b2i Q7 MK2b r?B+? r2
+M i?2M mb2 iQ /2}M2 i?2 bm#b+`BTib bbQ+Bi2/ rBi? M ``vX q2?HH /2}M2 Qm` 2MmK2`iBQM b 7QHHQrb,
ivT2aT2+B2b42MmK2`iBQMU- "- sVc
h?Bb /2}M2b bT2+BH ivT2 MK2/aT2+B2bi?i ?b 2t+iHv i?`22 TQbbB#H2 pHm2b--"M/sX q2 +M
i?2M mb2 i?Bb 2MmK2`iBQMb /BK2MbBQM BM M ``vb 7QHHQrb,
_2H*(aT2+B2b)c
aBM+2 i?2aT2+B2bivT2 ?b QMHv i?`22 TQbbB#H2 pHm2b- i?Bb K2Mb i?i i?2 p2+iQ`*?b 2t+iHv i?`22
+QKTQM2MibX q2 +M i?2M `272` iQ i?2 BM/BpB/mH +QKTQM2Mib Q7*b*(aT2+B2bX)-*(aT2+B2bX")M/
*(aT2+B2bXs)X
"2+mb2 Bi Bb rFr`/ iQ +QMbiMiHv T`2}t 2+? bT2+B2b MK2 rBi?aT2+B2b- r2 +M /2}M2 72r +QMp2MB2Mi
+QMbiMib b 7QHHQrb,
+QMbiMiaT2+B2b 4aT2+B2bXc
+QMbiMiaT2+B2b "4aT2+B2bX"c
+QMbiMiaT2+B2b s4aT2+B2bXsc
AM i?Bb rv- r2 +M MQr `272` iQ i?2 +QM+2Mi`iBQM Q7 bT2+B2bAb*()X SmHHBM; HH Q7 i?Bb iQ;2i?2` r2
+M `2T`2b2Mi Qm` +?2KB+H bvbi2K mbBM; 2MmK2`iBQMb b,
KQ/2H_2+iBQMbn1MmK]JQ/2HBM; +?2KB+H `2+iBQM rBi? 2MmKb]
ivT2aT2+B2b42MmK2`iBQMU
-
"-
sVc
_2H*(aT2+B2b)]aT2+B2b +QM+2Mi`iBQMb]c
T`K2i2`_2HF(j)4&yXR-yXR-Ry'c
+QMbiMiaT2+B2b 4aT2+B2bXc
+QMbiMiaT2+B2b "4aT2+B2bX"c
+QMbiMiaT2+B2b s4aT2+B2bXsc
BMBiBH2[miBQM
*()4RXyc
*(")4RXyc
*(s)4yXyc
2[miBQM
/2`U*()V4@F(R) *() *(")YF(k) *(s)c
/2`U*(")V4@F(R) *() *(")YF(k) *(s)@F(j) *(") *(s)c
/2`U*(s)V4F(R) *() *(")@F(k) *(s)@F(j) *(") *(s)c
2M/_2+iBQMbn1MmKc
Ryy *?Ti2` RX .2b+`B#BM; "2?pBQ`

111. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
*QM+HmbBQM
AM i?Bb +?Ti2`- r2 b?Qr2/ ?Qr b2i Q7 +?2KB+H 2[miBQMb +QmH/ #2 `2T`2b2Mi2/ rBi? M/ rBi?Qmi ``vbX
q2 HbQ /2KQMbi`i2/ ?Qr i?22MmK2`iBQMivT2 +M #2 mb2/ BM +QMDmM+iBQM rBi? ``vb iQ KF2 i?2
`2bmHiBM; 2[miBQMb KQ`2 `2/#H2 #v `2TH+BM; MmK2`B+ BM/B+2b rBi? MK2bX 6m`i?2`KQ`2- i?Bb b2+iBQM
HbQ /2KQMbi`i2/ ?Qr i?22MmK2`iBQMivT2 +M #2 mb2/ MQi QMHv iQ BM/2t i?2 ``v- #mi HbQ iQ /2}M2
QM2 Q` KQ`2 /BK2MbBQMb BM i?2 /2+H`iBQMX
RXjXk_2pB2r
``v .2+H`iBQMb
avMit
``v /2+H`iBQM bvMit Bb p2`v bBKTH2X h?2 bvMit Bb i?2 bK2 b 7Q` MQ`KH p`B#H2 /2+H`iBQM
2t+2Ti i?2 p`B#H2 MK2 b?QmH/ #2 7QHHQr2/ #v bm#b+`BTib iQ bT2+B7v i?2 bBx2 Q7 2+? /BK2MbBQM Q7 i?2
``vX h?2 ;2M2`H 7Q`K 7Q` M ``v /2+H`iBQM rQmH/ #2,
o`B#H2hvT2 p`LK2(/BKR- /BKk-XXX- /BKL)c
r?2`2o`B#H2hvT2Bb JQ/2HB+ ivT2 HBF2_2HQ`AMi2;2`-p`LK2Bb i?2 MK2 Q7 i?2 p`B#H2X
AMi2;2` aBx2b
LQ`KHHv- i?2 /BK2MbBQM bT2+B}+iBQMb `2 bBKTHv BMi2;2`b i?i BM/B+i2 i?2 bBx2 Q7 i?i /BK2MbBQMX 6Q`
2tKTH2,
RXjX o2+iQ`b M/ ``vb RyR

112. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
_2Ht(8)c
AM i?Bb +b2-tBb M ``v Q7 `2H pHm2/ MmK#2`b rBi? QMHv QM2 /BK2MbBQM Q7 bBx2 8X Ai Bb TQbbB#H2 iQ mb2
T`K2i2`b Q` +QMbiMib bT2+B7v i?2 bBx2 Q7 M ``v-2X;X-
T`K2i2`AMi2;2`/R48c
+QMbiMiAMi2;2`/k4kc
_2Ht(/R- /k)c
GBMF2/ .BK2MbBQMb
b r2 rBHH b22 b?Q`iHv r?2M r2 /Bb+mbb i?2 p`BQmb``v 6mM+iBQMbUT;2 Ry8V BM JQ/2HB+- r2 +M 2p2M
mb2 i?2bBx27mM+iBQM iQ bT2+B7v i?2 bBx2 Q7 QM2 ``v BM i2`Kb Q7 MQi?2` ``vX *QMbB/2` i?2 7QHHQrBM;,
_2Ht(8)c
_2Hv(bBx2Ut-RV)c
AM i?Bb +b2-vrBHH ?p2 QM2 /BK2MbBQM Q7 bBx2 8X h?2 mb2 Q7 i?2 7mM+iBQMbBx2Ut-RVrBHH `2im`M i?2 bBx2 Q7
/BK2MbBQM R Q7 i?2 ``vtX h?2`2 `2 KMv TTHB+iBQMb r?2`2 Bi Bb mb27mH iQ 2tT`2bb i?i i?2 /BK2MbBQMb
Q7 /Bz2`2Mi ``vb `2 `2Hi2/ BM i?Bb rv U2X;X- 2Mbm`BM; i?i ``vb `2 bBx2/ bm+? i?i QT2`iBQMb HBF2
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p2+iQ`UV
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am#i`+iBQM U@V
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7mM+iBQMSQHvMQKBH]*`2i2 ;2M2`B+ TQHvMQKBH 7`QK +Q277B+B2Mib]
BMTmi_2Ht ]AM/2T2M/2Mi p`B#H2]c
BMTmi_2H+(,)]SQHvMQKBH +Q277B+B2Mib]c
QmiTmi_2Hv]*QKTmi2/ TQHvMQKBH pHm2]c
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hQ p2`B7v i?i i?Bb 7mM+iBQM Bb rQ`FBM; T`QT2`Hv- H2i?b mb2 Bi BM KQ/2HX *QMbB/2` i?2 7QHHQrBM; JQ/2HB+
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129. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
KQ/2H1pHmiBQMh2biR]JQ/2H i?i 2pHmi2b TQHvMQKBH]
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2[miBQM
v74SQHvMQKBHUiBK2-&R-@k-k'Vc
vT4iBK2k@k iBK2Ykc
2M/1pHmiBQMh2biRc
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2pHmiBQM Q7 i?2 TQHvMQKBH-vT- rBi? QM2 +QKTmi2/ #v Qm` 7mM+iBQM-v7- r2 b22 i?2v `2 B/2MiB+H,
.Bz2`2MiBiBQM
Ai Bb +QKTH2i2Hv THmbB#H2 i?i i?Bb TQHvMQKBH 2pHmiBQM KB;?i #2 mb2/ iQ `2T`2b2Mi [mMiBiv i?i rb
mHiBKi2Hv /Bz2`2MiBi2/ #v i?2 JQ/2HB+ +QKTBH2`X h?2 7QHHQrBM; 2tKTH2b Bb /KBii2/Hv +QMi`Bp2/- #mi
Bi /2KQMbi`i2b ?Qr bm+? TQHvMQKBH KB;?i +QK2 iQ #2 /Bz2`2MiBi2/ BM KQ/2H,
KQ/2H.B772`2MiBiBQMR]JQ/2H i?i /B772`2MiBi2b 7mM+iBQM]
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2[miBQM
v74SQHvMQKBHUiBK2-&R-@k-k'Vc
vT4iBK2k@k iBK2Ykc
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2M/.B772`2MiBiBQMRc
RX9X 6mM+iBQMb RRN

130. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
>2`2 r2 ?p2 i?2 bK2 2[miBQMb 7Q`v7- 2pHmi2/ mbBM;SQHvMQKBH- M/vT- 2pHmi2/ /B`2+iHv b
TQHvMQKBHX "mi r2?p2 //2/ irQ //BiBQMH p`B#H2b-/nv7M//nvT`2T`2b2MiBM; i?2 /2`BpiBp2 Q7
v7M/vT- `2bT2+iBp2HvX A7 r2 ii2KTi iQ +QKTBH2 i?Bb KQ/2H i?2 +QKTBH2` Bb p2`v HBF2Hv iQ i?`Qr M
2``Q` `2Hi2/ iQ i?2 2[miBQM 7Q`/nv7X h?2 `2bQM Bb i?i Bi ?b MQ rv iQ +QKTmi2 i?2 /2`BpiBp2 Q7
v7X h?Bb Bb #2+mb2- mMHBF2vTr?B+? Bb +QKTmi2/ rBi? bBKTH2 2tT`2bbBQM- r2?p2 ?B//2M i?2 /2iBHb
Q7 ?Qrv7Bb +QKTmi2/ #2?BM/ i?2 7mM+iBQMSQHvMQKBHX AM ;2M2`H- JQ/2HB+ iQQHb /Q MQi HQQF i i?2
BKTH2K2MiiBQMb Q7 7mM+iBQMb iQ +QKTmi2 /2`BpiBp2b M/- 2p2M B7 i?2v /B/- /2i2`KBMBM; i?2 /2`BpiBp2 Q7
M `#Bi``v H;Q`Bi?K Bb MQi M 2bv i?BM; iQ /QX
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MMQiiBQMb
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iQ ;Bp2 i?2K ??BMib? #Qmi ?Qr iQ /2H rBi? +2`iBM bBimiBQMbX MMQiiBQMb `2 Hrvb ?QTiBQMH?
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/2}M2 M2r 2pHmiBQM 7mM+iBQM-SQHvMQKBHqBi?.2`BpiBp2- b 7QHHQrb,
7mM+iBQMSQHvMQKBHqBi?.2`BpiBp2
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BMTmi_2Ht ]AM/2T2M/2Mi p`B#H2]c
BMTmi_2H+(,)]SQHvMQKBH +Q277B+B2Mib]c
QmiTmi_2Hv]*QKTmi2/ TQHvMQKBH pHm2]c
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2M/SQHvMQKBHqBi?.2`BpiBp2c
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7mM+iBQMX q?i Bi BM/B+i2b Bb i?i i?2 7mM+iBQMSQHvMQKBH6B`bi.2`BpiBp2b?QmH/ #2 mb2/ iQ 2pHmi2
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biM/- Ki?2KiB+HHv- r?i Bb `2[mB`2/X _2+HH Qm` Q`B;BMH /2}MBiBQM Q7 Qm` TQHvMQKBH BMi2`TQHiBQM
7mM+iBQM,
p(x,⃗c)=
N
!
i=1
cix
N−i
LQi2 i?ipiF2b irQ `;mK2MibX A7 r2 rBb? iQ /Bz2`2MiBi2p#v bQK2 `#Bi``v p`B#H2z- r2 +M mb2
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Rky *?Ti2` RX .2b+`B#BM; "2?pBQ`

131. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
q2 +M /2`Bp2 i?2 7QHHQrBM; `2HiBQMb 7`QK Qm` Q`B;BMH /2}MBiBQM Q7pX 6B`bi- 7Q` i?2 T`iBH /2`BpiBp2 Q7
prBi? `2bT2+i iQxr2 ;2i,
∂p
∂x
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)
r?2`2c
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Bb /2}M2/ b,
c
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∂p
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⃗
di)
r?2`2 i?2p2+iQ`
⃗
diBb i?2i
th
+QHmKM Q7 MNxNB/2MiBiv Ki`BtX
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/x
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M/
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iQ T`QpB/2 7mM+iBQMb iQ 2pHmi2
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bdfX AM Qm` +b2- i?i 7mM+iBQM-SQHvMQKBH6B`bi.2`BpiBp2rQmH/ #2 /2}M2/ b 7QHHQrb,
7mM+iBQMSQHvMQKBH6B`bi.2`BpiBp2
]6B`bi /2`BpiBp2 Q7 i?2 7mM+iBQM SQHvMQKBH]
BMTmi_2Htc
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#v H2p2`;BM; Qm` TQHvMQKBH 2pHmiBQM 7mM+iBQMX
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KQ/2H.B772`2MiBiBQMk]JQ/2H i?i /B772`2MiBi2b 7mM+iBQM mbBM; /2`BpiBp2 MMQiiBQM]
_2Hv7c
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2[miBQM
v74SQHvMQKBHqBi?.2`BpiBp2UiBK2-&R-@k-k'Vc
vT4iBK2k@k iBK2Ykc
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2M/.B772`2MiBiBQMkc
RX9X 6mM+iBQMb RkR

132. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
aBKmHiBM; i?Bb KQ/2H M/ +QKT`BM; `2bmHib- r2 b22 ;`22K2Mi #2ir22Mv7M/vTb r2HH b/nv7M/
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AMi2`TQHiBQM
AM i?Bb +?Ti2`- r2 rBHH 2tKTH2 /Bz2`2Mi rvb Q7 BKTH2K2MiBM; bBKTH2 QM2 /BK2MbBQM BMi2`TQHiBQM
b+?2K2X q2?HH bi`i rBi? M TT`Q+? i?i Bb r`Bii2M +QKTH2i2Hv BM JQ/2HB+ M/ i?2M b?Qr M Hi2`MiBp2
BKTH2K2MiiBQM i?i +QK#BM2b JQ/2HB+ rBi? *X h?2 /pMi;2b M/ /Bb/pMi;2b Q7 2+? TT`Q+?
rBHH i?2M #2 /Bb+mbb2/X
JQ/2HB+ AKTH2K2MiiBQM
6mM+iBQM .2}MBiBQM
6Q` i?Bb 2tKTH2- r2 bbmK2 i?i r2 BMi2`TQHi2 /i BM i?2 7QHHQrBM; 7Q`K,
AM/2T2M/2Mi o`B#H2-x.2T2M/2Mi o`B#H2-y
x1 y1
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xn yn
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b?QmH/ `2im`M M BMi2`TQHi2/ pHm2 7Q`yX am+? 7mM+iBQM +QmH/ #2 BKTH2K2Mi2/ BM JQ/2HB+ b 7QHHQrb,
Rkk *?Ti2` RX .2b+`B#BM; "2?pBQ`

133. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
7mM+iBQMAMi2`TQHi2o2+iQ`]AMi2`TQHi2 7mM+iBQM /27BM2/ #v p2+iQ`]
BMTmi_2Ht ]AM/2T2M/2Mi p`B#H2]c
BMTmi_2Hv#`(,-k)]AMi2`TQHiBQM /i]c
QmiTmi_2Hv ].2T2M/2Mi p`B#H2]c
T`Qi2+i2/
AMi2;2`Bc
AMi2;2`M4bBx2Uv#`-RV]LmK#2` Q7 BMi2`TQHiBQM TQBMib]c
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H;Q`Bi?K
bb2`iUt=4v#`(R-R)-]AM/2T2M/2Mi p`B#H2 Kmbi #2 ;`2i2` i?M Q` 2[mH iQ]Yai`BM;Uv#`(R-
!→R)VVc
bb2`iUtI4v#`(M-R)-]AM/2T2M/2Mi p`B#H2 Kmbi #2 H2bb i?M Q` 2[mH iQ]Yai`BM;Uv#`(M-
!→R)VVc
B,4Rc
r?BH2t=4v#`(BYR-R)HQQT
B,4BYRc
2M/r?BH2c
T,4Ut@v#`(B-R)VfUv#`(BYR-R)@v#`(B-R)Vc
v,4T v#`(BYR-k)YUR@TV v#`(B-k)c
2M/AMi2`TQHi2o2+iQ`c
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BMTmi_2Ht ]AM/2T2M/2Mi p`B#H2]c
BMTmi_2Hv#`(,-k)]AMi2`TQHiBQM /i]c
QmiTmi_2Hv ].2T2M/2Mi p`B#H2]c
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Qm` 7mM+iBQM- i?2BMTmi`;mK2Miv#``2T`2b2Mib i?2 BMi2`TQHiBQM /i M/ i?2QmiTmi`;mK2Miv
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SQHvMQKBH 1pHmiBQMUT;2 RReV 2tKTH2- i?2b2 `2 2z2+iBp2Hv BMi2`K2/Bi2 p`B#H2b mb2/ BMi2`MHHv
#v i?2 7mM+iBQMX AM i?Bb +b2-BBb ;QBM; iQ #2 mb2/ b M BM/2t p`B#H2-MBb i?2 MmK#2` Q7 /i TQBMib
BM Qm` BMi2`TQHiBQM /i M/T`2T`2b2Mib r2B;?i mb2/ BM Qm` BMi2`TQHiBQM b+?2K2X
qBi? HH i?2 p`B#H2 /2+H`iBQMb Qmi Q7 i?2 rv- r2 +M MQr BKTH2K2Mi i?2H;Q`Bi?Kb2+iBQM Q7 Qm`
7mM+iBQM,
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!→R)VVc
bb2`iUtI4v#`(M-R)-]AM/2T2M/2Mi p`B#H2 Kmbi #2 H2bb i?M Q` 2[mH iQ]Yai`BM;Uv#`(M-
!→R)VVc
B,4Rc
r?BH2t=4v#`(BYR-R)HQQT
B,4BYRc
2M/r?BH2c
T,4Ut@v#`(B-R)VfUv#`(BYR-R)@v#`(B-R)Vc
v,4T v#`(BYR-k)YUR@TV v#`(B-k)c
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[x1,xn]X A7 MQi- M 2``Q` K2bb;2 rBHH #2 ;2M2`i2/ 2tTHBMBM; r?v i?2 BMi2`TQHiBQM 7BH2/X
h?2 `2bi Q7 i?2 7mM+iBQM b2`+?2b 7Q` i?2 pHm2 Q7Bbm+? i?ixi<=x<xi+1X PM+2 i?i pHm2 Q7B?b
RX9X 6mM+iBQMb Rkj

134. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
#22M B/2MiB}2/- i?2 BMi2`TQHi2/ pHm2 Bb +QKTmi2/ b bBKTHv,
y=p¯yi+1,2+(1−p)¯yi,2
r?2`2
p=
x−¯yi,1
¯yi+1,1−¯yi,1]
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−1
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150. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
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BMTmiAMTmihvT2R `;LK2R]/2b+`BTiBQM Q7 `;mK2MiR]c
XXX
BMTmiAMTmihvT2L `;LK2L,4/27mHioHm2L]/2b+`BTiBQM Q7 `;mK2MiL]c
QmiTmiPmiTmihvT2R `2im`MLK2R]/2b+`BTiBQM Q7 `2im`M pHm2 R]c
XXX
QmiTmiPmiTmihvT2L `2im`MLK2L]/2b+`BTiBQM Q7 `2im`M pHm2 L]c
T`Qi2+i2/
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XXX
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MMQiiBQMUF2vR4pHm2R-F2vk4pHm2kVc
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214. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
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215. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
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im4+ UT?Bn`2H@#fkVc
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217. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
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KQ/2HX AM i?Bb KQ/2H- r2 rBHH B;MQ`2 i?2 BM2`iB Q7 i?2 ;2` 2H2K2Mib- 2{+B2M+v HQbb2b BM i?2 ;2` M/
Mv #+FHb? i?i KB;?i 2tBbi #2ir22M i?2 i22i? BM i?2 ;2`X _2+HH Qm` /Bb+mbbBQM #Qmi.B;;BM; .22T2`
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KQ/2HX AMbi2/- r2 rBHH 7Q+mb QMHv QM i?2 `2HiBQMb?BT #2ir22M ;2` BMTmi bT22/ M/ QmiTmi bT22/X
AM ivTB+H bvbi2K /vMKB+b +Hbb- i?2 2[miBQMb i?i /2b+`B#2 i?2 #2?pBQ` Q7 ;2` `2 /2`Bp2/ b
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τaωa+τbωb=0
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i?2B` bmK Kmbi #2 x2`Q- bBM+2 Qm` ;2` KQ/2H /Q2b MQi BM+Hm/2 i?2 BM2`iB Q7 i?2 ;2` 2H2K2Mib M/-
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kXjX *QKTQM2Mib kyd

218. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
:Bp2M i?2b2 irQ 7+ib- r2 +M bm#biBimi2 i?2 `2HiBQMb?BT #2ir22M i?2 bT22/b BMiQ i?2 `2HiBQMb?BT #2ir22M
i?2 TQr2`b M/ ;2i,
τaRωb+τbωb=0
h?Bb HHQrb mb iQ +M+2H Qmiωb7`QK i?2 2[miBQM M/ `2``M;BM; i2`Kb ;Bp2b mb,
τb=−Rτa
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i?i i?2`2 Bb bQK2i?BM; KBbbBM; ?2`2X JQ`2 bT2+B}+HHv- i?2`2 Bb bQK2i?BM; BKTHB+BiHv bbmK2/ i?i Bb MQi
M2+2bb`BHv `2bQM#H2 bbmKTiBQMX
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!
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219. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
_2+iM;H2U
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/Q +QMbi`BM i?2 M;mH` TQbBiBQMb Q7 i?2 irQ b?7ibX 6m`i?2`KQ`2- i?2`2 Kv #2 bQK2 TTHB+iBQMb U2X;X
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+-
switch
n=10
1 2
transformer
D1
C=1e-2
capacitor
R=100load
step
0.25
D2
D3
D4
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gnd
p p_load
n_load
n=n
1 2
transformer
D1
C=C
capacitor
D2
D3
D4
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311. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
KQ/2H``v1tKTH2
SQBMi T(k)c
SQBMi [(k-j)c
o2+iQ`j. p(9)c
2[miBQM
TXt4&R-k'cff T(R)Xt 4 R- T(k)Xt 4 k
[(,-j)Xv4&9-8'cff [(R-j)Xv 4 9- [(k- j)Xv 4 8c
[Xt4(R-k-jc9-8-e)ff [(R-R)Xt 4 R-
ff [(R-k)Xt 4 k-
ff [(k-j)Xt 4 e
pXt(R)4&R-k-j-9'cff p(R)Xt(R) 4 R- p(k)Xt(R) 4 k-
ff p(j)Xt(R) 4 j- p(9)Xt(R) 4 9
p(,)Xt(R)4&R-k-j-9'cff p(R)Xt(R) 4 R- p(k)Xt(R) 4 k-
ff p(j)Xt(R) 4 j- p(9)Xt(R) 4 9
p(k,j)Xt(R)4&k-j'cff p(k)Xt(R) 4 k- p(j)Xt(R) 4 j
p(R)Xt4&R-k-j'cff p(R)Xt(R) 4 R- p(R)Xt(k) 4 k-
ff p(R)Xt(j) 4 j
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bm+? KQ/B}+iBQM Bb,
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ai2ToQHi;2 ob(8)c
b r2 br BM Qm` /Bb+mbbBQM Q7``vb Q7 *QKTQM2MiUT;2 jyyV- i?Bb Bb MQi QMHv H2;H JQ/2HB+- #mi
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T`K2i2`o7 pHm2- r2 ?p2 irQ +?QB+2bX h?2 }`bi Bb iQ bT2+B7v M ``v Q7 pHm2b-2X;X-
ai2ToQHi;2 ob(8)Uo74&k9-ke-k3-jy-jk'Vc
h?Bb bbB;Mb i?2 pHm2b BM i?2 p2+iQ`&k9-ke-k3-jy-jk'iQob(R)Xo7-ob(k)Xo7-ob(j)Xo7-ob(9)Xo7
M/ob(8)Xo7- `2bT2+iBp2HvX h?2 Qi?2` +?QB+2 r2 ?p2 Bb iQ ;Bp2 i?2 bK2 pHm2 iQ 2p2`v 2H2K2Mi BM i?2
``vX q2 +QmH/ mb2 i?Bb bK2 ``v BMBiBHBxiBQM bvMit-2X;X-
kX9X am#bvbi2Kb jyR

312. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
ai2ToQHi;2 ob(8)Uo74&k9-k9-k9-k9-k9'Vc
h?2 T`Q#H2K +QK2b r?2M i?2 MmK#2` Q7 2H2K2Mib BM M ``v Bb /2}M2/ #v T`K2i2`-2X;X-
T`K2i2`AMi2;2`Mc
ai2ToQHi;2 ob(M)Uo74f \\\ fVc
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hQ //`2bb i?Bb bBimiBQM- r2 +QmH/ mb2 i?2}HHUT;2 Ry8V 7mM+iBQM,
T`K2i2`AMi2;2`Mc
ai2ToQHi;2 ob(M)Uo747BHHUk9- MVVc
h?Bb Bb M ++2Ti#H2 bQHmiBQMX "mi BK;BM2 B7 r2 rMi2/ iQ KQ/B7v #Qi? i?2 pHm2 Q7o7M/ i?2KBM
ii`B#mi2 BMbB/2o7\ q2?/ 2M/ mT rBi? bQK2i?BM; HBF2 i?Bb,
T`K2i2`AMi2;2`Mc
ai2ToQHi;2 ob(M)Uo7UKBM47BHHUy-MVV47BHHUk9- MVVc
qBi? M2bi2/ KQ/B}+iBQMb- i?Bb FBM/ Q7 i?BM; +M ;2i +QKTHB+i2/ [mB+FHvX 6Q`imMi2Hv- JQ/2HB+ BM+Hm/2b
72im`2 iQ /2H rBi? bm+? bBimiBQMbX "v TH+BM; i?22+?F2vrQ`/ BM 7`QMi Q7 KQ/B}+iBQM- i?i
KQ/B}+iBQM Bb TTHB2/ iQ 2p2`v BMbiM+2-2X;X-
T`K2i2`AMi2;2`Mc
ai2ToQHi;2 ob(M)U2+?o7UKBM4yV4k9Vc
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S`QT;iBQM
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i?2M T`QT;i2b Q` +b+/2b /QrM iQ Bib +QKTQM2MibX 6Q` 2tKTH2- +QMbB/2` i?2 7QHHQrBM; bvbi2K KQ/2H
mb2/ BM Qm` /Bb+mbbBQM Q7"bB+ _QiiBQMH *QKTQM2MibUT;2 RNeV,
rBi?BMJQ/2HB+"v1tKTH2X*QKTQM2MibX_QiiBQMHX1tKTH2bc
KQ/2HaJ.
*QKTQM2MibX.KT2` /KT2`kU/4RVc
*QKTQM2MibX:`QmM/ ;`QmM/c
*QKTQM2MibXaT`BM; bT`BM;kU+48Vc
*QKTQM2MibXAM2`iB BM2`iBkUC4R-
T?BU7Bt2/4i`m2- bi`i4RV-
rU7Bt2/4i`m2- bi`i4yVVc
*QKTQM2MibX.KT2` /KT2`RU/4yXkVc
*QKTQM2MibXaT`BM; bT`BM;RU+4RRVc
*QKTQM2MibXAM2`iB BM2`iBRU
C4yX9-
T?BU7Bt2/4i`m2- bi`i4yV-
rU7Bt2/4i`m2- bi`i4yVVc
2[miBQM
ff XXX
2M/aJ.c
A7 r2 rMi2/ iQ mb2 i?Bb KQ/2H BM /Bz2`2Mi +QMi2tib r?2`2 i?2 pHm2b Q7 i?2 +QKTQM2Mi T`K2i2`b- HBF2
/- KB;?i p`v- r2 +QmH/ KF2/ T`K2i2` i i?2 bm#bvbi2K H2p2H M/ i?2M T`QT;i2 Bi /QrM BMiQ i?2
?B2``+?v mbBM; KQ/B}+iBQMX h?2 `2bmHi rQmH/ HQQF bQK2i?BM; HBF2 i?Bb,
rBi?BMJQ/2HB+"v1tKTH2X*QKTQM2MibX_QiiBQMHX1tKTH2bc
KQ/2HaJ.
jyk *?Ti2` kX P#D2+i@P`B2Mi2/ JQ/2HBM;

313. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
BKTQ`iJQ/2HB+XaAmMBibX c
T`K2i2`_QiiBQMH.KTBM;*QMbiMi /c
*QKTQM2MibX.KT2` /KT2`kU/4/Vc
ff XXX
h?2`2 Bb QM2 +QKTHB+iBQM ?2`2X Ai Bb TQbbB#H2 7Q` mb2` iQ +QK2 HQM; M/ +?M;2 i?2 pHm2 Q7/KT2`kX
/BMbi2/ Q7 KQ/B7vBM; i?2/T`K2i2` BM i?2aJ.KQ/2HX hQ pQB/ ?pBM; i?2/T`K2i2` M/ i?2
/KT2`kX/T`K2i2` 7`QK ;2iiBM; Qmi Q7 bvM+ U?pBM; /Bz2`2Mi pHm2bV- r2 +M T2`KM2MiHv #BM/ i?2K
mbBM; i?27BMH[mHB}2`,
rBi?BMJQ/2HB+"v1tKTH2X*QKTQM2MibX_QiiBQMHX1tKTH2bc
KQ/2HaJ.
BKTQ`iJQ/2HB+XaAmMBibX c
T`K2i2`_QiiBQMH.KTBM;*QMbiMi /c
*QKTQM2MibX.KT2` /KT2`kU7BMH/4/Vc
ff XXX
"v //BM; i?27BMH[mHB}2`- r2 `2 BM/B+iBM; i?i Bi Bb MQ HQM;2` TQbbB#H2 iQ KQ/B7v i?2 pHm2 Q7
/KT2`kX/X Mv KQ/B}+iBQMb Kmbi #2 K/2 iQ/QMHvX
:BpBM; HH Q7 i?2 ??`/@rB`2/? MmK2`B+H pHm2b BM i?2aJ.KQ/2H i?2 bK2 i`2iK2Mi- r2 rQmH/ 2M/ mT
rBi? ?B;?Hv `2mb#H2 KQ/2H HBF2 i?Bb,
rBi?BMJQ/2HB+"v1tKTH2X*QKTQM2MibX_QiiBQMHX1tKTH2bc
KQ/2HaJ.
BKTQ`iJQ/2HB+XaAmMBibX c
T`K2i2`_QiiBQMH.KTBM;*QMbiMi /R- /kc
T`K2i2`_QiiBQMHaT`BM;*QMbiMi +R- +kc
T`K2i2`AM2`iB CR- Ckc
T`K2i2`M;H2 T?BRnBMBi4y- T?BknBMBi4yc
T`K2i2`M;mH`o2HQ+Biv rRnBMBi4y- rknBMBi4yc
*QKTQM2MibX.KT2` /KT2`kU7BMH/4/kVc
*QKTQM2MibX:`QmM/ ;`QmM/c
*QKTQM2MibXaT`BM; bT`BM;kU7BMH+4+kVc
*QKTQM2MibXAM2`iB BM2`iBkU
7BMHC4Ck-
T?BU7Bt2/4i`m2-7BMHbi`i4T?BknBMBiV-
rU7Bt2/4i`m2-7BMHbi`i4rknBMBiVVc
*QKTQM2MibX.KT2` /KT2`RU7BMH/4/RVc
*QKTQM2MibXaT`BM; bT`BM;RU7BMH+4+RVc
*QKTQM2MibXAM2`iB BM2`iBRU
7BMHC4CR-
T?BU7Bt2/4i`m2-7BMHbi`i4T?BRnBMBiV-
rU7Bt2/4i`m2-7BMHbi`i4rRnBMBiVVc
2[miBQM
ff XXX
2M/aJ.c
A7 r2 rMi2/ iQ mb2 bT2+B}+ b2i Q7 T`K2i2` pHm2b- r2 +QmH/ /Q Bi BM QM2 Q7 irQ rvbX PM2 rv rQmH/
#2 iQ 2ti2M/ i?2 T`K2i2`Bx2/ KQ/2H #Qp2 M/ BM+Hm/2 KQ/B}+iBQM BM i?22ti2M/bbii2K2Mi-2X;X-
KQ/2HaT2+B7B+aJ.
2ti2M/baJ.U/k4R- +k48- Ck4R-
/R4yX8- +R4RR- CR4yX9-
T?BRnBMBi4RVc
LQi2 i?i r2 /B/ MQi M22/ iQ BM+Hm/2 KQ/B}+iBQMb 7Q` i?2 pHm2b Q7T?BknBMBi-rRnBMBiM/rknBMBi-
bBM+2 i?Qb2 T`K2i2`b r2`2 /2+H`2/ rBi? /27mHi pHm2bX AM ;2M2`H-/27mHi pHm2b 7Q` T`K2i2`b
b?QmH/ QMHv #2 mb2/ r?2M i?Qb2 /27mHib `2 `2bQM#H2 7Q` i?2 pbi KDQ`Biv Q7 +b2bX h?2
`2bQM 7Q` i?Bb Bb i?i B7 T`K2i2` ?b MQ /27mHi pHm2 KQbi JQ/2HB+ +QKTBH2`b rBHH ;2M2`i2 r`MBM;
kX9X am#bvbi2Kb jyj

314. JQ/2HB+ #v 1tKTH2- _2H2b2 pyXeXy
H2`iBM; vQm i?i pHm2 Bb `2[mB`2/X "mi B7 /27mHi pHm2 Bb i?2`2- Bi rBHH bBH2MiHv mb2 i?2 /27mHi pHm2X
A7 i?i /27mHi pHm2 Bb MQi `2bQM#H2 Q` ivTB+H- i?2M vQm rBHH bBH2MiHv BMi`Q/m+2 M mM`2bQM#H2 pHm2
BMiQ vQm` KQ/2HX
"mi `2im`MBM; iQ i?2 iQTB+ Q7 T`QT;iBQM- i?2 Qi?2` TT`Q+? i?i +QmH/ #2 mb2/ rQmH/ #2 iQ BMbiMiBi2
M BMbiM+2 Q7 i?2aJ.KQ/2H M/ mb2 KQ/B}+iBQMb QM i?2 /2+H`2/ p`B#H2 iQ bT2+B7v T`K2i2` pHm2b-
2X;X-
aJ. KvbK/U/k4R- +k48- Ck4R-
/R4yX8- +R4RR- CR4yX9-
T?BRnBMBi4RVc
q2?HH /272` i?2 /Bb+mbbBQM QM r?B+? Q7 i?2b2 TT`Q+?2b Bb #2ii2` mMiBH i?2 mT+QKBM; +?Ti2` QM`+?B@
i2+im`2bUT;2 jy9VX
kX8`+?Bi2+im`2b
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h?i- #v Bib2H7- Bb p2`v BMi2`2biBM; #2+mb2 r2 r2`2 #H2 iQ pQB/ ?pBM; iQ rQ``v #Qmi ?Qr r2 rQmH/
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