Về điểm tới hạn của một lớp hệ phương trình vi phân tuyến tính và á tuyến tính

✣❸■ ❍➴❈ ✣⑨ ◆➂◆●
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼
❑❍❖❆ ❚❖⑩◆

❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P
✣➋ ❚⑨■✿
❱➋ ✣■➎▼ ❚❰■ ❍❸◆ ❈Õ❆ ▼❐❚ ▲❰P ❍➏ P❍×❒◆● ❚❘➐◆❍
❱■ P❍❹◆ ❚❯❨➌◆ ❚➑◆❍ ❱⑨ ⑩ ❚❯❨➌◆ ❚➑◆❍

●✐→♦ ✈✐➯♥ ữợ

r

tỹ

ỗ

ợ







ử ử
ớ õ



ỵ ồ t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹

✵✳✷ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺

✵✳✸ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺

✵✳✹ ✣è✐ t÷đ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺

✵✳✺ ❈➜✉ tró❝ ❧✉➟♥ ✈➠♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻

✶ ▼ët sè ❦❤→✐ ♥✐➺♠ ♠ð ✤➛✉
✶✳✶

✼

❍➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼

✶✳✶✳✶

❑❤→✐ ♥✐➺♠ ❝❤✉♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼

✶✳✶✳✷

❍➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❤➺ sè ❤➡♥❣ ✳ ✳

✽

✶✳✷

❍➺ → t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽

✶✳✸

❍➺ ♦t♦♥♦♠ ✭❍➺ tü ✤✐➲✉ ❦❤✐➸♥✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽

✶✳✹

✣✐➸♠ tỵ✐ ❤↕♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

ỵ ✤✐➸♠ tỵ✐ ❤↕♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶

✶✳✺✳✶

❚✉②➳♥ t➼♥❤ ❤â❛ t↕✐ ✤✐➸♠ tỵ✐ ❤↕♥ ✳ ✳ ✳ ✳





ỵ ỹ ê♥ ✤à♥❤ ❝õ❛ ❤➺ t✉②➳♥ t➼♥❤✮ ✳ ✳ ✳ ✳





ỵ ỹ ờ ừ t✉②➳♥ t➼♥❤✮ ✳ ✳ ✳ ✳ ✳

✶✾

✶✳✻ ▼ët ✈➔✐ ♣❤÷ì♥❣ ♣❤→♣ sè ❣✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✳ ✳ ✳

✷✶

✶✳✻✳✶

P❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷

✷✷


✶✳✻✳✷

P❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡✲❑✉tt❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✷

✷ Ù♥❣ ❞ö♥❣ ♣❤➛♥ tt ởt số trữ
ừ ởt ợ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ✈➔ →
t✉②➳♥ t➼♥❤
✷✳✶

✷✹

❱➲ ♠ët sè ✤➦❝ tr÷♥❣ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t


ử ỵ ✤✐➸♠ tỵ✐ ❤↕♥
(0, 0) ✈➔ ♥â✐ rã ♥â ❧➔ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥✱ ê♥ ✤à♥❤ ❤❛②

❦❤æ♥❣ ê♥ ✤à♥❤✳ ❑✐➸♠ tr❛ ỗ t




ử ỵ ợ tợ (x0 , y0 )

✤➸ ♣❤➙♥ ❧♦↕✐ ✈➔ ①➨t t➼♥❤ ê♥ tr
ỗ t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✳✷

✸✵

❱➲ ♠ët sè ✤➦❝ tr÷♥❣ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ → t✉②➳♥
t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✷✳✶

✸✼

❳→❝ ✤à♥❤ ❧♦↕✐ ✤✐➸♠ tỵ✐ ❤↕♥ (0, 0) ❝õ❛ ❤➺ → t✉②➳♥ t➼♥❤✱
♠✐➯✉ t↔ sü ①➜♣ ①➾ ✤à❛ ữỡ
tợ t ý ✤÷đ❝ ①→❝ ✤à♥❤ tr♦♥❣ ↔♥❤ ♣❤❛✳ ✳

✷✳✷✳✷

✸✼

❳→❝ ✤à♥❤ ❝→❝ ✤✐➸♠ tợ ừ ữ r ự
t ê♥ ✤à♥❤ ❝õ❛ tø♥❣ ✤✐➸♠✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✽

❙ü ♣❤➙♥ ♥❤→♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✸

✷✳✸ Ù♥❣ ❞ö♥❣✿ ❙ü sè♥❣ sât ❝õ❛ ♠ët ❧♦➔✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✺

✷✳✷✳✸

❑➳t ❧✉➟♥

✺✵

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✺✶

P❤ö ❧ö❝

✺✷

✸


ớ õ
ỵ ồ t
ồ õ ✈à tr➼ r➜t q✉❛♥ trå♥❣ ✤è✐ ✈ỵ✐ ❝→❝ ♠ỉ♥ ❦❤♦❛ ❤å❝ ❦❤→❝ ✈➔
tr♦♥❣ ✤í✐ sè♥❣✳ ◗✉❛ ♥â ❝â t❤➸ ❣✐↔✐ q✉②➳t ♥❤✐➲✉ ✈➜♥ ✤➲ ✤è✐ ✈ỵ✐ ❝→❝ ♠ỉ♥ ❤å❝
t➼♥❤ t♦→♥✱ ❣✐↔✐ t❤➼❝❤ ❝→❝ q✉② ❧✉➟t ♣❤→t tr✐➸♥ tr♦♥❣ tü ♥❤✐➯♥ ✈➔ ✤í✐ sè♥❣ ❝♦♥
♥❣÷í✐ t❤ỉ♥❣ q✉❛ ✈✐➺❝ ♠✐➯✉ t↔ ✈➔ ①➙② ❞ü♥❣ ❝→❝ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥✳ ❚r♦♥❣
t❤ü❝ t➳ ♥❤✐➲✉ ❜➔✐ t♦→♥ t❛ ❝â t❤➸ ❜✐➸✉ ❞✐➵♥ q✉❛ ♠ët ❤➺ ♣❤÷ì♥❣ t➻♥❤ ✈✐ ♣❤➙♥
t✉②➳♥ t➼♥❤ ❤♦➦❝ → t✉②➳♥ t➼♥❤✱ ✤➦❝ t tr t ỵ s ồ ổ
q ♥❣❤✐➯♥ ❝ù✉ t➼♥❤ ❝❤➜t ❝õ❛ ❤➺ ♥➔② t❛ ❝â t❤➸ ♠✐➯✉ t↔ ♠ët ❝→❝❤ ❝ö

t❤➸ ❝→❝ t➼♥❤ ❝❤➜t✱ ✤➦❝ ✤✐➸♠ ❝õ❛ ✈➜♥ ✤➲ ✈➔ ❣✐↔✐ q✉②➳t ✤÷đ❝ ✈➜♥ ✤➲ ✤÷đ❝

tợ ừ ữỡ tr t t➼♥❤ ✈➔ → t✉②➳♥ t➼♥❤
❝â ✈à tr➼ q✉❛♥ trå♥❣ ✈➔ ✤â♥❣ ✈❛✐ trá t❤❡♥ ❝❤èt tr♦♥❣ ✈✐➺❝ ♠æ t↔ ❝→❝
ừ tổ q ỗ t ừ õ ❚❤➜② ✤÷đ❝ ✈❛✐ trá q✉❛♥ trå♥❣
✤â ✈➔ ✤÷đ❝ sü ✤ë♥❣ ủ ỵ ừ t ữợ ❚❙✳ ▲➯ ❍↔✐
❚r✉♥❣ ♥➯♥ tæ✐ ❝❤å♥ ✤➲ t➔✐ ❱➲ ✤✐➸♠ tợ ừ ởt ợ ữỡ

tr t t➼♥❤ ✈➔ → t✉②➳♥ t➼♥❤ ❧➔♠ ❧✉➟♥ ✈➠♥ tèt ♥❣❤✐➺♣✳
❑➳t ❤đ♣ ✈ỵ✐ ✈✐➺❝ sû ❞ư♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ✺✳✷ ✤➸ ①➙② ❞ü♥❣ tr÷í♥❣
✈➨❝ tì ❝õ❛ ❤➺ ✈➔ ❣✐↔✐ ❝→❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❤➺ sè ❤➡♥❣
❜➡♥❣ ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ sè ✤➸ t❤➜② ✤÷đ❝ rã ♥➨t ❞→♥❣ ✤✐➺✉✱ q✉ÿ ✤↕♦ ❝õ❛

✹


❝→❝ ♥❣❤✐➺♠ tr♦♥❣ ❤➺ ✤➣ ❝❤♦✳ ✣➙② ❧➔ ♠ët ❝æ♥❣ ❝ư ❦❤→ ♠↕♥❤ ✤÷đ❝ ❞ị♥❣
♥❤✐➲✉ tr♦♥❣ ❦ÿ t❤✉➟t ♥❤÷♥❣ tr♦♥❣ t♦→♥ ❤å❝ t❤➻ ♥â ❝á♥ ❦❤→ ♠ỵ✐ ♠➫✳ ❚❤ỉ♥❣
q✉❛ ✈✐➺❝ sû ❞ư♥❣ ♣❤➛♥ ♠➲♠ ♥➔② t❤➻ ❝ỉ♥❣ ✈✐➺❝ t➼♥❤ t♦→♥ ❝õ❛ ❝❤ó♥❣ t❛ s➩
♥❤❛♥❤ ❝❤â♥❣ ✈➔ ✤ì♥ ❣✐↔♥ ❤ì♥✳

✵✳✷ ▼ư❝ ự
r ỡ s ự ỵ sü ê♥ ✤à♥❤ ❝õ❛ ❤➺ t✉②➳♥ t➼♥❤ ✈➔ ♥❣❤✐➯♥
❝ù✉ ✤à♥❤ ỵ sỹ ờ ừ t t t❤ỉ♥❣ q✉❛ ❝→❝ ✈➼ ❞ư ❝ư t❤➸
✤➸ ♣❤→❝ ❤å❛ rã ♥➨t ❝→❝ ✤➦❝ tr÷♥❣ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤
✈➔ → t✉②➳♥ t➼♥❤✳

✵✳✸ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉
◆❣❤✐➯♥ ❝ù✉ ❝→❝ t➔✐ ❧✐➺✉ ❧✐➯♥ q✉❛♥ ✤➳♥ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥
t➼♥❤ ✈➔ → t✉②➳♥ t➼♥❤✱ ❝→❝ t➔✐ ❧✐➺✉ ✈➲ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ✺✳✷ ✤➸ ❣✐↔✐

q✉②➳t ❝→❝ ✈➼ ❞ö ❝ö t❤➸✳

✵✳✹ ✣è✐ t÷đ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉
◆❣❤✐➯♥ ❝ù✉ ✤✐➸♠ tợ ừ ữỡ tr t t ✈➔ →
t✉②➳♥ t➼♥❤ ❤❛✐ ❜✐➳♥ t❤ỉ♥❣ q✉❛ ❝→❝ ✈➼ ❞ư ❝ö t❤➸✳
❳➙② ❞ü♥❣ ❝→❝ ❝➙✉ ❧➺♥❤ tr♦♥❣ ♠❛t❤❡♠❛t✐❝❛ ✺✳✷ ✤➸ ✈➩ tr÷í♥❣ ✈➨❝ tì ✈➔
❣✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❜➡♥❣ ♠ët ✈➔✐ ♣❤÷ì♥❣ ♣❤→♣ sè✳

✺


✵✳✺ ❈➜✉ tró❝ ❧✉➟♥ ✈➠♥
◆❣♦➔✐ ♣❤➛♥ ♠ð ✤➛✉✱ ❦➳t ❧✉➟♥✱ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✈➔ ♣❤➛♥ ♣❤ö ❧ö❝ tr♦♥❣
❧✉➟♥ ✈➠♥ ỗ õ ữỡ s
ữỡ ởt số ♠ð ✤➛✉✳
❈❤÷ì♥❣ ✷✿ Ù♥❣ ❞ư♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ❝❤♦ ♠ët sè ✤➦❝ tr÷♥❣ ❝õ❛
❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ✈➔ → t✉②➳♥ t➼♥❤✳

▲❮■ ❈❷▼ ❒◆
❚→❝ ❣✐↔ ①✐♥ t❤➸ ❤✐➺♥ ỏ t ỡ s s tợ t ữợ
r õ ỳ ủ ỵ õ õ qỵ tr q tr tỹ
t ỗ t❤í✐ t→❝ ❣✐↔ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ ✤➳♥ ❝→❝
t❤➛②✱ ❝ỉ tr♦♥❣ ❑❤♦❛ ❚♦→♥✱ ❚r÷í♥❣ ✣↕✐ ❍å❝ ❙÷ P❤↕♠✲ ✣↕✐ ❍å❝ ✣➔ ◆➤♥❣
✤➣ ✤ë♥❣ ✈✐➯♥ ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤✳

✻


❈❤÷ì♥❣ ✶
▼ët sè ❦❤→✐ ♥✐➺♠ ♠ð ✤➛✉

✶✳✶ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤
✶✳✶✳✶

❑❤→✐ ♥✐➺♠ ❝❤✉♥❣

❍➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝â ❞↕♥❣✿









dy1
dx

= a11 (x)y1 + a12 (x)y2 + ... + a1n (x)yn + g1 (x),

dy2
dx

= a21 (x)y1 + a22 (x)y2 + ... + a2n (x)yn + g2 (x),



..............................................................................





 dyn = a (x)y + a (x)y + ... + a (x)y + g (x),
n1
1
n2
2
nn
n
n
dx

✭✶✳✶✮

tr♦♥❣ ✤â x ❧➔ ❜✐➳♥ ✤ë❝ ❧➟♣✱ y1 , y2 , ..., yn ❧➔ ❝→❝ ❤➔♠ ➞♥ ❝➛♥ t➻♠ ✈➔ a11 (x), a12 (x), ...,
ann (x) ❧➔ ❝→❝ ❤➺ sè ❤♦➦❝ ❝→❝ ❤➺ sè tü ❞♦ ❝õ❛ ❤➺✱ g1 (x), g2 (x), ..., gn (x) ❧➔

số trữợ õ tr ữủ ❣✐↔ t❤✐➳t ❧✐➯♥ tö❝ tr➯♥ ❦❤♦↔♥❣
I = (a; b) ⊂ R

ũ ỵ tr t õ t t ữợ
y = A(x)y + g(x),

tr õ y =

T

y1 y2 ... yn

✱ A(x) = (aij (x))n×n ✱ y =

T

g(x) =

g1 (x) g2 (x) ... gn (x)

✼

✳

✭✶✳✷✮
T

y1 y2 ... yn

✱


◆➳✉ g(x) ≡ (0) t❤➻ ❤➺ ✭✶✳✷✮ ❧➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ t❤✉➛♥
♥❤➜t✳

✶✳✶✳✷

❍➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❤➺ sè ❤➡♥❣

❍➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❤➺ sè ❤➡♥❣ ❝â ❞↕♥❣✿
✭✶✳✸✮

y = Ay + g(x),

tr♦♥❣ ✤â y =

T

y1 y2 ... yn

✱ A = (aij )n×n ✱ y =
T

g(x) =

g1 (x) g2 (x) ... gn (x)

T

y1 y2 ... yn

✱

✳ ◆➳✉ g(x) ≡ (0) t❤➻ ❤➺ ✭✶✳✸✮ ❧➔ ❤➺

♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ t❤✉➛♥ ♥❤➜t ❤➺ sè ❤➡♥❣✳

✶✳✷ ❍➺ → t✉②➳♥ t➼♥❤
❍➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝â ❞↕♥❣✿



dx
dt

= ax + by + r(x, y),



dy
dt

= cx + dy + s(x, y),

✭✶✳✹✮

✤÷đ❝ ❣å✐ ❧➔ ❤➺ → t✉②➳♥ t➼♥❤✱ tr♦♥❣ ✤â t ❧➔ ❜✐➳♥ ✤ë❝ ❧➟♣✱ x = x(t), y = y(t)
❧➔ ❝→❝ ❤➔♠ ➞♥ ❝➛♥ t➻♠✱ r(x, y), s(x, y) ❧➔ ❝→❝ ❤➔♠ t❤❡♦ ❜✐➳♥ x, y ✳ ❈→❝ ❤➔♠
✤÷đ❝ ❣✐↔ t❤✐➳t ❧✐➯♥ tư❝ tr➯♥ ❦❤♦↔♥❣ I = (a, b) ⊂ R ✳

✶✳✸ ❍➺ ♦t♦♥♦♠ ✭❍➺ tü ✤✐➲✉ ❦❤✐➸♥✮
❍➺ ♦t♦♥♦♠ ✭❍➺ tü ✤✐➲✉ ❦❤✐➸♥✮ ❝â ❞↕♥❣✿



dx
dt

= f (x, y),



dy
dt

= g(x, y),

✽

✭✶✳✺✮


tr♦♥❣ ✤â t ❧➔ ❜✐➳♥ ✤ë❝ ❧➟♣✱ x = x(t), y = y(t) ❧➔ ❝→❝ ❤➔♠ ➞♥ ❝➛♥ t➻♠✱
f (x, y), g(x, y) ❧➔ ❝→❝ ❤➔♠ t❤❡♦ ❜✐➳♥ x, y ✈➔ ❝❤ó♥❣ ❦❤↔ ✈✐ ❧✐➯♥ tư❝ tr♦♥❣

♠✐➲♥ R ❝õ❛ ♠➦t ♣❤➥♥❣ Oxy ✲ ♠➦t ♣❤➥♥❣ ♣❤❛ ❝õ❛ ❤➺ ♦t♦♥♦♠✳

✶✳✹ ✣✐➸♠ tỵ✐ ❤↕♥
✣à♥❤ ♥❣❤➽❛✳
❈❤♦ ❤➺ ♦t♦♥♦♠ ✭❤➺ tü ✤✐➲✉ ❦❤✐➸♥✮



dx
dt

= f (x, y),



dy
dt

= g(x, y),

✭✶✳✻✮


 f (x0 , y0 ) = 0,
(x0 , y0 ) ữủ ồ tợ ❤↕♥ ❝æ ❧➟♣ ♥➳✉ t❤ä❛
 g(x , y ) = 0.
0

0

t ừ tợ
ã út tợ ❤↕♥ (x0 , y0 ) ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ ♥ót ♥➳✉ t❤ä❛ ♠➣♥ ✷

✤✐➲✉ ❦✐➺♥✿

✕ ▼å✐ q✉ÿ ✤↕♦ ✤➲✉ t✐➳♥ tỵ✐ (x0 , y0 ) ❦❤✐ t → +∞ ❤♦➦❝ ♠å✐ q✉ÿ ✤↕♦
✤➲✉ rí✐ ①❛ (x0 , y0 ) ❦❤✐ t → +∞✳

✕ ▼å✐ q✉ÿ ✤↕♦ ✤➲✉ t✐➳♣ ①ó❝ ✈ỵ✐ ✤÷í♥❣ t❤➥♥❣ ✤✐ q✉❛ (x0 , y0 ) t↕✐
(x0 , y0 )✳

▼ët ✤✐➸♠ ♥ót ♥❤÷ tr➯♥ ✤÷đ❝ ❣å✐ ❧➔ ♥ót ❝❤➼♥❤ t❤÷í♥❣ ❤❛② ♥ót ❝❤➼♥❤✳
◆➳✉ ♠å✐ q✉ÿ ✤↕♦ ✤➲✉ ✤✐ ✤➳♥ ✤✐➸♠ ♥ót t❤➻ ♥â ✤÷đ❝ ❣å✐ ❧➔ ♥ót ❧ã♠✳
◆➳✉ ♠å✐ q✉ÿ ✤↕♦ ✤➲✉ ❧ị✐ ①❛ ✤✐➸♠ ♥ót t❤➻ ♥â ữủ ồ út ỗ
ã s ự ♠é✐ ❝➦♣ q✉ÿ ✤↕♦ ✤è✐ ❞✐➺♥ ❦❤→❝ ♥❤❛✉ ❦❤æ♥❣ ❝â

❝➦♣ t ú ợ ữớ t q tợ ❤↕♥ t❤➻ ✤✐➸♠ ♥ót
❝❤➼♥❤ ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ ❤➻♥❤ s❛♦ ❤❛② ✤✐➸♠ s❛♦✳

✾


• ◆ót ♣❤✐ ❝❤➼♥❤✿ ▼å✐ q✉ÿ ✤↕♦ trø r❛ ♠ët q ố t

ú ợ ởt ữớ t❤➥♥❣ ✤✐ q✉❛ ✤✐➸♠ tỵ✐ ❤↕♥ t❤➻ ♥ót ✤â ❣å✐ út

ã ỹ tợ ữủ ❣å✐ ❧➔ ✤✐➸♠ ②➯♥ ♥❣ü❛ ♥➳✉ ♠å✐ q✉ÿ

✤↕♦ ❧➔ ❜→♥ trö❝ ❝õ❛ ❤②♣❡❜♦❧✱ ✤✐➸♠ (x(t), y(t)) ❞➛♥ ✤➳♥ (x0 , y0 ) t❤❡♦
trư❝ Ox ✈➔ rí✐ ①❛ t❤❡♦ trư❝ Oy ❦❤✐ t → +∞ ✈➔ ❝â ❤❛✐ q✉ÿ ✤↕♦ ❞➛♥
✤➳♥ (x0 , y0 ) ♥❤÷♥❣ ✤➲✉ ❦❤ỉ♥❣ ❜à ❝❤➦♥ ❦❤✐ t +
ã tợ ờ tợ ❤↕♥ ✤÷đ❝ ❣å✐ ❧➔ ê♥ ✤à♥❤ ♥➳✉ ❦❤✐ ✤✐➸♠

✤➛✉ (x1 , y1 ) ✤õ ❣➛♥ (x0 , y0 ) t❤➻ ✤✐➸♠ (x(t), y(t)) ❧✉ỉ♥ ❣➛♥ (x0 , y0 ) ✈ỵ✐
♠å✐ t > 0✳
❍❛② ✤➦t X(t) = (x(t), y(t))✱ X0 = (x0 , y0 )✱ X1 = (x1 , y1 ) ❦❤✐ ✤â✱ X0 ❧➔
✤✐➸♠ ê♥ ✤à♥❤ ♥➳✉ ∀ε > 0, ∃δ > 0 : |X0 − X1 | < δ t❤➻ |X0 − X(t)| <
ε, ∀t > 0✳

✣✐➸♠ tỵ✐ ❤↕♥ ❦❤æ♥❣ ê♥ ✤à♥❤ ♥➳✉ ♥â ❦❤æ♥❣ ❧➔ ✤✐➸♠ ê♥ ✤à♥❤ tù❝ ❧➔∃ε >
0, ∃δ > 0 : |X0 − X1 | < δ ♠➔ |X0 − X(t)| < ε, ∀t > 0✳

◆ót ❧ã♠ ❝â |X0 − X(t)| < ε, ∀t > 0 ❞♦ ✤â ♥â ❧➔ ✤✐➸♠ ♥ót ê♥ ✤à♥❤✳
• tợ ữủ ồ t õ tợ ờ

ữủ q q✉ÿ ✤↕♦ ❦➼♥✱ t✉➛♥ ❤♦➔♥✳
❙✉② r❛ ♠å✐ t➙♠ ✤➲✉ ê♥ ✤à♥❤✳
• ❚✐➺♠ ❝➟♥ ê♥ ✤à♥❤ ✭ê♥ ✤à♥❤ t✐➺♠ ❝➟♥✮✿ ✣✐➸♠ tợ (x0 , y0 ) ữủ ồ

ờ t✐➺♠ ❝➟♥ ♥➳✉ ❧➔ ✤✐➸♠ ê♥ ✤à♥❤ ✈➔ ♠å✐ q✉ÿ ✤↕♦ ✤õ ❣➛♥ (x0 , y0 )
✤➲✉ t✐➳♥ tỵ✐ (x0 , y0 ) ❦❤✐ t → +∞✳
❍❛② ∃δ > 0 : |X0 − X1 | < δ, lim X(t) = X0 ✳
t→+∞

✶✵


• ✣✐➸♠ ê♥ ✤à♥❤ ①♦➢♥ ✭✣✐➸♠ ①♦➢♥ ❧ã♠✮✿ ✣✐➸♠ tỵ✐ ❤↕♥ ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠

ê♥ ✤à♥❤ ①♦➢♥ ♥➳✉ ♥â ❧➔ ♠ët ✤✐➸♠ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ ♠➔ ❝→❝ q✉ÿ ✤↕♦
❝❤✉②➸♥ ✤ë♥❣ ①♦➢♥ è❝ q✉❛♥❤ ♥â ✈➔ t✐➳♥ ✤➳♥ ♥â✳ ◆➳✉ ❝→❝ q✉ÿ ✤↕♦ ❝❤✉②➸♥
✤ë♥❣ ①❛ ❞➛♥ ♥â t❤➻ ♥â ❣å✐ ổ ờ ỗ

ỵ tợ


t õ t tợ ❤↕♥

❈❤♦ ❤➺ ♦t♦♥♦♠ ✭❍➺ tü ✤✐➲✉ ❦❤✐➸♥✮✿



dx
dt

= f (x, y),



dy
dt

= g(x, y),

✭✶✳✼✮

❦❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t r➡♥❣ ✤✐➸♠ tỵ✐ ❤↕♥ ❝ỉ ❧➟♣ ❧➔
(x0 , y0 ) = (0, 0) ♥➳✉ ❦❤æ♥❣ t❤ü❝ ❤✐➺♥ ♣❤➨♣ ✤ê✐ ❜✐➳♥✿ u = x − x0 , v = y − y0 ✳

❑❤✐ ✤â✱

dx
dt

=

du dy
dt ✱ dt

= dv
dt ♥➯♥ ❤➺ tữỡ ữỡ ợ

dx = f (u + x0 , v + y0 ) = f1 (u, v),
dt



dy
dt

= g(u + x0 , v + y0 ) = g1 (u, v).

✭✶✳✽✮

❚❛ ❝â ♥❤➟♥ ①➨t ❝→❝ ✤÷í♥❣ ❝♦♥❣ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ✭✶✳✼✮ tr♦♥❣ ♠➦t ♣❤➥♥❣
Oxy ✤÷đ❝ s✉② r❛ ❜➡♥❣ ❝→❝❤ tà♥❤ t✐➳♥ (u, v) → (u + x0 , v + y0 ) ❝õ❛ ✤÷í♥❣

❝♦♥❣ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ✭✶✳✽✮ tr♦♥❣ ♠➦t ♣❤➥♥❣ uv ✳ ❱➻ ✈➟② tr♦♥❣ ❧➙♥ ❝➟♥ ❝õ❛
❤❛✐ tợ tữỡ ự (x0 , y0 ) tr ♠➦t ♣❤➥♥❣ xy ✈➔ (0, 0) tr♦♥❣ ♠➦t
♣❤➥♥❣ uv ❤❛✐ ↔♥❤ ♣❤❛ trỉ♥❣ ❤♦➔♥ t♦➔♥ ♥❤÷ ♥❤❛✉✳
◆❣♦➔✐ r❛ t❛ ❝â t❤➸ ✤÷❛ ❤➺ ♦t♦♥♦♠ ✈➲ ❤➺ → t✉②➳♥ t➼♥❤✳ ❚❛ ❝â ❤➔♠ f (x, y)
❧➔ ❦❤↔ ✈✐ ❧✐➯♥ tö❝ q✉❛♥❤ ✤✐➸♠ ❝è ✤à♥❤ (x0 , y0 )✱ →♣ ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ ❚❛②❧♦r
❝❤♦ ❤➔♠ ❤❛✐ ❜✐➳♥ ✤÷đ❝✿
f (x0 + u, y0 + v) = f (x0 , y0 ) + fx (x0 , y0 )u + fy (x0 , y0 )v + r(u, v)

✶✶


tr♦♥❣ ✤â ♣❤➛♥ ❞÷ r(u, v) t❤ä❛

r(u,v)
√
2
2
(u,v)→(0,0) u +v

lim

= 0✳

⑩♣ ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ ❚❛②❧♦r ❝❤♦ ❤➔♠ f, g tr♦♥❣ ❤➺ ✭✶✳✽✮ ✈ỵ✐ ❣✐↔ t❤✐➳t
(x0 , y0 ) ❧➔ ✤✐➸♠ tỵ✐ ❤↕♥ ❝æ ❧➟♣ t❛ ❝â✿

 du = fx (x0 , y0 )u + fy (x0 , y0 )v + r(u, v),
dt



dv
dt

= gx (x0 , y0 )u + gy (x0 , y0 )v + s(u, v),

✭✶✳✾✮

tr♦♥❣ ✤â r(u, v), s(u, v) ❧➛♥ ❧÷đt ❧➔ ♣❤➛♥ ❞÷ ❝õ❛ ❝→❝ ❤➔♠ f, g t❤ä❛✿
r(u, v)
s(u, v)
√
= lim √
= 0,
(u,v)→(0,0) u2 + v 2
(u,v)→(0,0) u2 + v 2
lim

✈➔ ❦❤✐ ❣✐→ trà ❝õ❛ u ✈➔ v ♥❤ä t❤➻ ❝→❝ ♣❤➛♥ ❞÷ r(u, v) ✈➔ s(u, v) ❧➔ r➜t ♥❤ä
✭♥❤ä ✤➳♥ ♥é✐ ❝â t❤➸ s♦ s→♥❤ ✈ỵ✐ u ✈➔ v ✮✳
❉♦ ✤â ❦❤✐ (u, v) → (0, 0) t❤➻ ❤➺ ✭✶✳✾✮ ①➜♣ ①➾ ✈ỵ✐ ❤➺ t✉②➳♥ t➼♥❤✿




du
dt

= fx (x0 , y0 )u + fy (x0 , y0 )v,



dv
dt

= gx (x0 , y0 )u + gy (x0 , y0 )v.

✭✶✳✶✵✮

❑❤✐ ✤â sü t✉②➳♥ t➼♥❤ ❤â❛ ❝õ❛ ❤➺ ✭✶✳✼✮ t↕✐ ✤✐➸♠ tỵ✐ ❤↕♥ (x0 , y0 ) ❧➔ ❤➺
T

✭✶✳✶✵✮ ❤❛② ❤➺ u = Ju tr♦♥❣ ✤â u =

u v

✱ ♠❛ tr➟♥ ❤➺ sè ❝õ❛ ♥â ✤÷đ❝

❣å✐ ❧➔ ♠❛ tr➟♥ ❏❛❝♦❜✐❛♥✿

J(x0 , y0 ) = 

fx (x0 , y0 ) fy (x0 , y0 )

gx (x0 , y0 ) gy (x0 , y0 )




ừ f g ữợ ữủ t (x0 , y0 )



ỵ ỹ ê♥ ✤à♥❤ ❝õ❛ ❤➺ t✉②➳♥ t➼♥❤✮

✣➸ ♥❣❤✐➯♥ ❝ù✉ ✤✐➸♠ tỵ✐ ❤↕♥ (0, 0) ❝õ❛ ❤➺ t✉②➳♥ t➼♥❤ t❛ ❝â t❤➸ ũ
ữỡ tr r ố ợ ữỡ tr



x
y





=

a b
c d

x
y


,


ợ tr số ữủ ỵ

ỵ 1 , 2 trà r✐➯♥❣ ❝õ❛ ♠❛ tr➟♥ ❤➺ sè ❆ ❝õ❛ ❤➺
t✉②➳♥ t➼♥❤ ❤❛✐ ❝❤✐➲✉✿




dx
dt

= ax + by,



dy
dt

= cx + dy,

✭✶✳✶✶✮

✈ỵ✐ ad − bc = 0✱ ❑❤✐ ✤â t❛ ❝â✿

●✐→ trà r✐➯♥❣ λ , λ

✣✐➸♠ tỵ✐ ❤↕♥ (0; 0)

●❤✐ ❝❤ó

λ1 = λ2 > 0

út ổ ờ

ỗ

1 = λ2 < 0

◆ót ♣❤✐ ❝❤➼♥❤✱ t✐➺♠ ❝➟♥ ê♥ ✤à♥❤✳

▲➔ ✤➾♥❤ ❝❤➻♠✳

λ1 < 0 < λ2

✣✐➸♠ ②➯♥ ♥❣ü❛✱ ❦❤æ♥❣ ê♥ ✤à♥❤✳

λ1 = λ2 < 0

◆ót ❝❤➼♥❤ ❤♦➦❝ ♣❤✐ ❝❤➼♥❤✱ t✐➺♠ ❝➟♥ ê♥ ✤à♥❤✳

λ1 = λ2 > 0

◆ót ❝❤➼♥❤ ❤♦➦❝ ♣❤✐ ❝❤➼♥❤✱ ❦❤æ♥❣ ê♥ ✤à♥❤✳

λ1,2 = p ± qi (p > 0)

✣✐➸♠ ố ổ ờ

ỗ ố

1,2 = p ± qi (p < 0)

✣✐➸♠ ①♦➢♥ è❝✱ t✐➺♠ ❝➟♥ ê♥ ✤à♥❤✳

▲➔ ❧ã♠ ①♦➢♥ è❝✳

λ1,2 = ±qi

❚➙♠✱ ê♥ ✤à♥❤ ♥❤÷♥❣ ❦❤ỉ♥❣ t✐➺♠ ❝➟♥ ê♥ ✤à♥❤✳

1

2

❇↔♥❣ ✶✳✶✿ ❇↔♥❣ ❤➺ t❤è♥❣ ❝→❝ ❧♦↕✐ ✤✐➸♠ tỵ✐ ❤↕♥ ❝õ❛ ❤➺ t✉②➳♥ t➼♥❤✳
◆❤➟♥ ①➨t✿ ❑❤✐ ❝→❝ ❤➺ sè ❝â sü ①→♦ trë♥ ♥❤ä t❤➻ ❞➝♥ ✤➳♥ sü ①→♦ trë♥ ♥❤ä
❝õ❛ λ1 , λ2 ❦❤✐ ✤â✿
• ◆➳✉ ❞➜✉ ❝õ❛ Re(λi )✱ i = 1, 2 ❦❤æ♥❣ ✤ê✐ t tợ ổ

ờ

ã 1,2 = qi t à1,2 = r si t tợ ❤↕♥ ❧➔ t➙♠ t❤➔♥❤

✤✐➸♠ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ ♥➳✉ r < 0✱ ❦❤ỉ♥❣ ê♥ ✤à♥❤ ♥➳✉ r > 0✳
• ◆➳✉ 1 = 2 t à1,2 R t tợ ❤↕♥ ❧➔ ♥ót ✈➝♥ ❧➔ ♥ót✱ t❤➔♥❤
µ1,2 ♣❤ù❝ ❧✐➯♥ ❤đ♣ t❤➻ ✤✐➸♠✱ tr♦♥❣ ❝→❝ tr÷í♥❣ ❤đ♣ t❤➻ t➼♥❤ ê♥ ✤à♥❤

❦❤ỉ♥❣ ✤ê✐✳

✶✸


❈❤ù♥❣ ♠✐♥❤✳
❍➺ t✉②➳♥ t➼♥❤✿






x



=

y

a b
c d




x
y


,

✈ỵ✐ ♠❛ tr➟♥ ❤➺ sè ❤➡♥❣ ❆ ❝â ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣✿
det(A − λI) =

a−λ

b

c

d−λ

= (a − λ)(d − λ) − bc = 0.

❱➻ ad − bc = 0 ♥➯♥ ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❧✉ỉ♥ tỗ t
ã tr r tỹ ♥❤÷♥❣ ❝ị♥❣ ❞➜✉ ✭λ1 = λ2 > 0 ❤♦➦❝
λ1 = λ2 < 0✮✿

❚❤❡♦ ❣✐↔ t❤✐➳t t❛ ❝â ♠❛ tr➟♥ ❆ ❝â ❝→❝ ✈➨❝ tì r✐➯♥❣ v1 , v2 ♣❤ư t❤✉ë❝
t✉②➳♥ t➼♥❤ ✈➔ ♥❣❤✐➺♠ tê♥❣ q✉→t x(t) =

T

x(t) y(t)

❝õ❛ ❤➺ ✭✶✳✶✶✮

❝â ❞↕♥❣✿
x(t) = c1 v1 eλ1 t + c2 v2 eλ2 t .

✭✶✳✶✷✮

❍➻♥❤ ✶✳✶✿ ❍➺ tå❛ ✤ë ①✐➯♥ uv ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ✈➨❝ tì r✐➯♥❣ v1, v2
❚r♦♥❣ ❤➺ tå❛ ✤ë uv ✭tr♦♥❣ ✤â trư❝ u ✈➔ v ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ❝→❝
✈➨❝ tì r✐➯♥❣ v1 ✈➔ v2 ✮ ❤➔♠ u(t) ✈➔ v(t) ❝õ❛ ✤✐➸♠ ❝❤✉②➸♥ ✤ë♥❣ x(t) ❧➔

✶✹


tứ õ t ữợ s s ợ ❝→❝ ✈➨❝ tì v1 ✈➔ v2 ✳ ❉♦
✤â t❤❡♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✷✮ q✉ÿ ✤↕♦ ❝õ❛ ❤➺ ✭✶✳✶✶✮ ✤÷đ❝ ♠✐➯✉ t↔ ❜ð✐✿
u(t) = u0 eλ1 t , v(t) = v0 eλ2 t ,

✭✶✳✶✸✮

tr♦♥❣ ✤â u0 = u(0) ✈➔ v0 = v(0)✳
◆➳✉ v0 = 0 t❤➻ q✉ÿ ✤↕♦ ♥➡♠ tr➯♥ trö❝ u✱ t÷ì♥❣ tü ♥➳✉ u0 = 0 t❤➻
q✉ÿ ✤↕♦ ♥➡♠ tr➯♥ trư❝ v ✳
◆➳✉ u0 ✈➔ v0 ✤➲✉ ❦❤→❝ ❦❤ỉ♥❣ t❤➻ ✤÷í♥❣ ❝♦♥❣ t❤❛♠ sè ✭✶✳✶✸✮ ❝â t❤➸
✈✐➳t t❤➔♥❤ v = Cuk tr♦♥❣ ✤â k =

λ1

λ2

> 0✳ ❈→❝ ✤÷í♥❣ ❝♦♥❣ ♥❣❤✐➺♠ t✐➳♣

①ó❝ t↕✐ (0, 0) ✈ỵ✐ trư❝ u ♥➳✉ k > 1✱ ✈ỵ✐ trư❝ v ♥➳✉ 0 < k < 1✳
❑❤✐ õ t ữủ tợ (0, 0) ởt ♥ót ♣❤✐ ❝❤➼♥❤✳
◆➳✉ λ1 , λ2 > 0 t❤➻ ❝→❝ ✤÷í♥❣ ❝♦♥❣ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ✭✶✳✶✷✮ ✈➔ ✭✶✳✶✸✮
❧➺❝❤ ✈➲ ❣è❝ t t (0, 0) ỗ
λ1 , λ2 < 0 t❤➻ ❝→❝ ✤÷í♥❣ ❝♦♥❣ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ✭✶✳✶✷✮ ✈➔ ✭✶✳✶✸✮
s→t ❣➛♥ ❣è❝ ❦❤✐ t t➠♥❣✱ ✈➻ ✈➟② (0, 0) ❧➔ ✤➾♥❤ ❝❤➻♠✳
• ●✐→ trà r✐➯♥❣ t❤ü❝ ❦❤→❝ ♥❤❛✉ ♥❤÷♥❣ tr→✐ ❞➜✉ ✭λ1 < 0 < λ2 ✮✿

❚÷ì♥❣ tü tr÷í♥❣ ❤đ♣ ❣✐→ trà r✐➯♥❣ t❤ü❝ ❝ị♥❣ ❞➜✉ ♥❤÷♥❣ λ1 < 0 < λ2 ✳
❑❤✐ ✤â✿

✕ ◗✉ÿ ✤↕♦ ✈ỵ✐ u0 = 0 ❤♦➦❝ v0 = 0 ♥➡♠ tr➯♥ trư❝ u ✈➔ v ✤✐ q✉❛ ✤✐➸♠
tỵ✐ ❤↕♥ (0, 0)✳

✕ ◗✉ÿ ✤↕♦ ✈ỵ✐ u0 ✈➔ v0 ❦❤→❝ ✵ ❧➔ ♥❤ú♥❣ ✤÷í♥❣ ❝♦♥❣ ❝õ❛ ❞↕♥❣ v =
Cuk tr♦♥❣ ✤â k =

λ1
λ2

< 0✳ ❱➻ ✈➟② q✉ÿ ✤↕♦ ♣❤✐ t✉②➳♥ ❣✐è♥❣ ♥❤÷

❤②♣❡❜♦❧✱ ✤✐➸♠ tỵ✐ ❤↕♥ (0, 0) ❧➔ ♠ët ✤✐➸♠ ②➯♥ ♥❣ü❛ ❦❤ỉ♥❣ ê♥ ✤à♥❤✳
• ●✐→ trà r✐➯♥❣ ❧➔ ❤❛✐ ♥❣❤✐➺♠ t❤ü❝ ❜➡♥❣ ♥❤❛✉ ✭λ1 = λ2 ∈ R✮✿

✶✺

❚❛ ①➨t tr÷í♥❣ ❤đ♣ λ = λ1 = λ2 = 0✱ ❦❤✐ ✤â t➼♥❤ ❝❤➜t ❝õ❛ ✤✐➸♠ tỵ✐
❤↕♥ (0, 0) ♣❤ư t❤✉ë❝ ✈➔♦ t➼♥❤ ✤ë❝ ❧➟♣ ❝õ❛ ❝→❝ ✈➨❝ tì r✐➯♥❣ ❝õ❛ ♠❛ tr➟♥
❆✳
❚❛ ❝â q✉ÿ ✤↕♦ ❝õ❛ ❤➺ ✭✶✳✶✶✮ tr♦♥❣ ❤➺ tå❛ ✤ë ①✐➯♥ uv ✤÷đ❝ ♠ỉ t↔✿
u(t) = u0 eλt , v(t) = v0 eλt .

◆➳✉ k =

λ1
λ2

= 1 t❤➻ q✉ÿ ✤↕♦ u0 = 0 ❝â ❞↕♥❣ v = Cv ❞♦ ✤â ✤➲✉ ♥➡♠

tr➯♥ ✤÷í♥❣ t❤➥♥❣ ✤✐ q✉❛ ❣è❝✳ ❱➻ ✈➟② (0, 0) ❧➔ ✤✐➸♠ ❤➻♥❤ s❛♦✳ ◆â
ỗ > 0 ó < 0✳
◆➳✉ ❝→❝ ❣✐→ trà r✐➯♥❣ λ = 0 ❝â ❝ò♥❣ tỡ r v1 t tỗ t ởt
tỡ r tê♥❣ q✉→t v2 s❛♦ ❝❤♦ (A − λI)v2 = v1 ✱ ❤➺ t✉②➳♥ t➼♥❤ x = Ax
❝â ✷ ♥❣❤✐➺♠ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✿
x1 (t) = v1 eλt , x2 (t) = (v1 t + v2 )eλt .

✭✶✳✶✹✮

❚ø ✭✶✳✶✹✮ s✉② r❛✱ tr♦♥❣ ❤➺ tå❛ ✤ë ①✐➯♥ v1 v2 ❤➔♠ tå❛ ✤ë u(t) ✈➔ v(t)
❝õ❛ ✤✐➸♠ ❝❤✉②➸♥ ✤ë♥❣ x(t) tr➯♥ q✉ÿ ✤↕♦
u(t) = (u0 + v0 t)eλt , v(t) = v0 eλt ,

tr♦♥❣ ✤â u0 = u(0), v0 = v(0)✳

✕ ◆➳✉ v0 = 0 t❤➻ q✉ÿ ✤↕♦ ♥➔② ♥➡♠ tr➯♥ trö❝ u✳
✕ ◆➳✉ v0 = 0 t❛ ❝â ♠ët q✉ÿ ✤↕♦ ♣❤✐ t✉②➳♥
dv
=
du

❚❛ ❝â

dv
dt
du
dt

λv0 eλt
λv0
=
=
.
λt
λt
v0 e + λ(u0 + v0 t)e
v0 + λ(u0 + v0 t)

dv t→+∞
−−−→
du −

0✳ ❙✉② r❛ q✉ÿ ✤↕♦ t✐➳♣ ①ó❝ ✈ỵ✐ trư❝ u✳ ❉♦ ✤â (0, 0)

❧➔ ♥ót ♣❤✐ ❝❤➼♥❤✳

◆➳✉ λ < 0 t❤➻ (0, 0) ❧➔ ♥ót ó
> 0 t (0, 0) út ỗ




• ●✐→ trà r✐➯♥❣ ❧➔ ♥❣❤✐➺♠ ♣❤ù❝ ❧✐➯♥ ❤ñ♣ ✭λ1,2 = p ± qi✮✿

●✐↔ sû ♠❛ tr➟♥ ❆ ❝â ❣✐→ trà r✐➯♥❣ λ = p + qi✱ λ¯ = p − qi ✈ỵ✐ p, q = 0
❝â ❝→❝ ✈➨❝ tì r✐➯♥❣ t÷ì♥❣ ù♥❣ v = a + bi✱ v¯ = a − bi✳ ❑❤✐ ✤â✱ ❤➺ t✉②➳♥
t➼♥❤ x = Ax ❝â ❤❛✐ ♥❣❤✐➺♠ t❤ü❝ ♣❤ö t❤✉ë❝✿
x1 (t) = ept (a cos(qt) − b sin(qt)), x2 (t) = ept (b cos(qt) + a sin(qt)),

✭✶✳✶✺✮
❑❤✐ ✤â✱ ❝→❝ t❤➔♥❤ ♣❤➛♥ x(t), y(t) ❝õ❛ ♥❣❤✐➺♠ x(t) = c1 x1 (t) + c2 x2 (t)
❞❛♦ ✤ë♥❣ ❣✐ú❛ ❣✐→ trà ➙♠ ✈➔ ❞÷ì♥❣ ❦❤✐ t t➠♥❣✳ ❉♦ ✤â ✤✐➸♠ tỵ✐ ❤↕♥
(0, 0) ❧➔ ♠ët ✤✐➸♠ ①♦➢♥ è❝✳
t→+∞

◆➳✉ ♣❤➛♥ t❤ü❝ p < 0 t❤➻ x(t) −−−−→ 0✱ s✉② r❛ ✤✐➸♠ (0, 0) ❧➔ ❧ã♠
①♦➢♥ è❝✳
◆➳✉ ♣❤➛♥ t❤ü❝ p > 0 t (0, 0) ỗ è❝✳
• ●✐→ trà r✐➯♥❣ ❧➔ ❤❛✐ ♥❣❤✐➺♠ t❤✉➛♥ ↔♦ ✭λ1,2 = ±qi✮✿

●✐↔ sû ♠❛ tr➟♥ ❆ ❝â ❣✐→ trà r✐➯♥❣ λ = qi, λ¯ = −qi ✈ỵ✐ ❝→❝ ✈➨❝ tì
r✐➯♥❣ t÷ì♥❣ ù♥❣ v = a + bi, v¯ = a − bi✳ ⑩♣ ❞ư♥❣ ✭✶✳✶✺✮ ✈ỵ✐ p = 0 ❤➺
t✉②➳♥ t➼♥❤ x = Ax ❝â ♥❣❤✐➺♠ ♣❤ö t❤✉ë❝✿
x1 (t) = a cos(qt) − b sin(qt), x2 (t) = b cos(qt) + a sin(qt).

❙✉② r❛ ♥❣❤✐➺♠ x(t) = c1 x1 (t) + c2 x2 (t) ❜✐➸✉ ❞✐➵♥ ♠ët ❡❧✐♣ ❝â t➙♠ ❧➔

(0, 0) tr♦♥❣ ♠➦t ♣❤➥♥❣ xy ✱ ❞♦ ✤â (0, 0) ❧➔ t➙♠ ê♥ ✤à♥❤✳

❱➼ ❞ö ✶✳✺✳✷✳✶✳ ▼❛ tr➟♥✿


A=



3
1 7
,
8 −3 −17

✶✼


❝â ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣✿
det(A−λI) =

7
8

3
8

−λ
− 38

17

8

7
17
9
= ( −λ)( −λ)+ = 0 ⇔ λ2 −3λ+2 = 0.
8
8
64
−λ

❙✉② r❛ ❤❛✐ ❣✐→ trà r✐➯♥❣ λ1 = 1 ✈➔ λ2 = 2 ✈ỵ✐ ❝→❝ ✈➨❝ tì r✐➯♥❣ t÷ì♥❣ ù♥❣
T

v1 = 3 1

✈➔ v2 = 1 3

T

✳

❍➻♥❤ ✶✳✷✿ ◆ót ❧ã♠ ♣❤✐ ❝❤➼♥❤ tr♦♥❣ ✈➼ ❞ö ✶✳✺✳✷✳✶
❚❛ ❝â ❜✐➸✉ ❞✐➵♥ q✉ÿ ✤↕♦ ❝õ❛ ❤➺ t✉②➳♥ t➼♥❤ x = Ax ♥❤÷ ❤➻♥❤ ✶✳✷✳ ❚❛
t❤➜② ❤❛✐ ✈➨❝ tỡ r t ữợ ừ q t t t➜t ❝↔ ❝→❝
q✉ÿ ✤↕♦ ❦❤→❝ ❤❛✐ trö❝ ❝õ❛ ❤➺ tå❛ ✤ë ①✐➯♥ ✤➲✉ t✐➳♣ ①ó❝ ✈ỵ✐ ♠ët trư❝ ①✐➯♥
✤✐ q✉❛ ❣è❝ tå❛ ✤ë✳
●✐→ trà r✐➯♥❣ ❧➔ ❤❛✐ ♥❣❤✐➺♠ t❤ü❝ ❦❤→❝ ữ ũ ữỡ
tợ (0, 0) út õ ỗ

ử ✶✳✺✳✷✳✷✳ ▼❛ tr➟♥✿


A=



1 −10 15
,
4 −15 8

❝â ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣✿
det(A−λI) =

− 25 − λ
− 15
4

15
4

5
225
1 145
= (− −λ)(2−λ)+
= 0 ⇔ λ2 + λ+
= 0.
2
16
2

16
2−λ

✶✽


❙✉② r❛ ❤❛✐ ❣✐→ trà r✐➯♥❣ λ1 = − 14 + 3i ✈➔ λ2 = − 14 − 3i✳ ❚❛ ❝â ❜✐➸✉ ❞✐➵♥ q✉ÿ

❍➻♥❤ ✶✳✸✿ ◆ót ❧ã♠ ①♦➢♥ è❝ ð ✈➼ ❞ö ✶✳✺✳✷✳✷
✤↕♦ ❝õ❛ ❤➺ t✉②➳♥ t➼♥❤ x = Ax ♥❤÷ ❤➻♥❤ ✶✳✸✳ ❚❛ ✤÷đ❝ ♠ët q✉ÿ ✤↕♦ ①♦➢♥
è❝ ✤✐➸♥ ❤➻♥❤ ❣➛♥ ✈➲ ❣è❝ ❦❤✐ t → +∞✳
●✐→ trà r✐➯♥❣ ❧➔ ❤❛✐ ♥❣❤✐➺♠ ♣❤ù❝ ❧✐➯♥ ❤ñ♣ ❝â ♣❤➛♥ t❤ü❝ ➙♠ ♥➯♥ ✤✐➸♠ tỵ✐
❤↕♥ (0, 0) ❧➔ ❧ã♠ ①♦➢♥ è❝ ✈➔ õ t ờ



ỵ ỹ ờ ✤à♥❤ ❝õ❛ ❤➺ → t✉②➳♥ t➼♥❤✮

❈❤♦ ❤➺ → t✉②➳♥ t➼♥❤✿



dx
dt

= ax + by + r(x, y),



dy

dt

= cx + dy + s(x, y),

✭✶✳✶✻✮

❝â (0, 0) ❧➔ ♠ët ✤✐➸♠ tỵ✐ ❤↕♥ ❝ỉ ợ ad bc = 0 ỵ ự tọ
t ờ ừ tợ ữ ❝õ❛ ❝→❝ sè ❤↕♥❣ ♣❤✐ t✉②➳♥
r(x, y), s(x, y) ❧➔ tữỡ ữỡ ợ sỹ trở ọ ừ số ừ

t t tữỡ ự

ỵ λ1 , λ2 ❧➔ ❝→❝ ❣✐→ trà r✐➯♥❣ ❝õ❛ ♠❛ tr số ừ
t t tữỡ ự ợ ❤➺ → t✉②➳♥ t➼♥❤ ✭✶✳✶✻✮✱ ✤✐➸♠ tỵ✐ ❤↕♥ (0, 0)✳
❑❤✐ ✤â✿

✶✾


●✐→ trà r✐➯♥❣ λ , λ ❝õ❛ ❤➺ t✉②➳♥ t➼♥❤ ❤â❛ ✣✐➸♠ tỵ✐ ❤↕♥ (0; 0) ❝õ❛ ❤➺ → t✉②➳♥ t➼♥❤
1

2

λ1 = λ2 > 0

◆ót ♣❤✐ ❝❤➼♥❤✱ ❦❤ỉ♥❣ ê♥ ✤à♥❤✳

λ1 = λ2 < 0

◆ót ♣❤✐ ❝❤➼♥❤✱ ê♥ ✤à♥❤✳

λ1 < 0 < λ2

✣✐➸♠ ②➯♥ ♥❣ü❛✱ ❦❤æ♥❣ ê♥ ✤à♥❤✳

λ1 = λ2 < 0

◆ót ê♥ ✤à♥❤ ❤♦➦❝ ✤✐➸♠ ①♦➢♥ è❝✳

λ1 = λ2 > 0

◆ót ❦❤ỉ♥❣ ê♥ ✤à♥❤ ❤♦➦❝ ✤✐➸♠ ①♦➢♥ è❝✳

λ1,2 = p ± qi (p > 0)

✣✐➸♠ ①♦➢♥ è❝✱ ❦❤æ♥❣ ê♥ ✤à♥❤✳

λ1,2 = p ± qi (p < 0)

✣✐➸♠ ①♦➢♥ è❝✱ t✐➺♠ ❝➟♥ ê♥ ✤à♥❤✳

λ1,2 = ±qi

❚➙♠ ❤♦➦❝ ✤✐➸♠ ①♦➢♥ è❝✱ ê♥ ✤à♥❤ ❤♦➦❝ ❦❤æ♥❣✳

❇↔♥❣ ✶✳✷✿ ❇↔♥❣ ❤➺ t❤è♥❣ ❝→❝ ❧♦↕✐ ✤✐➸♠ tỵ✐ ❤↕♥ ❝õ❛ ❤➺ → t✉②➳♥ t➼♥❤✳
❱➼ ❞ư ✶✳✺✳✸✳✶✳ ❳→❝ ✤à♥❤ ❧♦↕✐ ✈➔ sü ê♥ ✤à♥❤ ❝õ❛ ✤✐➸♠ tỵ✐ ❤↕♥ (4, 3) ❝õ❛
❤➺ → t✉②➳♥ t➼♥❤✿




dx
dt



= −10x − 3y + x2 + 33,

dy
dt

= 6x + 2y − xy − 18.

✭✶✳✶✼✮

❚❛ ❝â f (x, y) = −10x − 3y + x2 + 33✱ g(x, y) = 6x + 2y − xy − 18 ✈➔
x0 = 4✱ y0 = 3✳ ❙✉② r❛✿


 
−10 + 2x −3
fx (x, y) fy (x, y)
,
=
J(x, y) = 
gx (x, y) gy (x, y)
6−y
2−x



✈➻ ✈➟② J(4, 3) = 

−2 −3
3

−2


✳

❙✉② r❛ ❤➺ t✉②➳♥ t➼♥❤ t÷ì♥❣ ù♥❣ ❝õ❛ ❤➺ ✭✶✳✶✼✮✿




du
dt
dv
dt

= −2u − 3v,
= 3u − 2v.

✷✵

✭✶✳✶✽✮

P❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❝õ❛ ❤➺ ✭✶✳✶✽✮✿
−2 − λ

−3

3

−2 − λ

= 0 ⇔ (−2 − λ)2 + 9 = 0.

●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t❛ ♥❤➟♥ ✤÷đ❝ λ1 = −2 + 3i ✈➔ λ2 = −2 − 3i✳ ❙✉② r❛
✤✐➸♠ tỵ✐ ❤↕♥ (0, 0) ❝õ❛ ❤➺ ✭✶✳✶✽✮ ❧➔ ✤✐➸♠ ①♦➢♥ è❝✱ t✐➺♠ ❝➟♥ ê♥ ✤à♥❤✱ ❞♦ ✤â
✤✐➸♠ tỵ✐ ❤↕♥ (4, 3) ❝õ❛ ❤➺ ✭✶✳✶✼✮ ❧➔ ✤✐➸♠ ①♦➢♥ è❝✱ t✐➺♠ ❝➟♥ ê♥ ✤à♥❤✳

❍➻♥❤ ✶✳✹✿ ◗✉ÿ ✤↕♦ ①♦➢♥ è❝ ❝õ❛ ❤➺ ✭✶✳✶✽✮

❷♥❤ ♣❤❛ ❝õ❛ ❤➺ ✭✶✳✶✼✮

✶✳✻ ▼ët ✈➔✐ ♣❤÷ì♥❣ ♣❤→♣ sè ❣✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐
♣❤➙♥
❚❛ ✤✐ ❣✐↔✐ q✉②➳t ❜➔✐ t♦→♥✿
x = f (t, x, y),

x(t0 ) = x0 ,

y = g(t, x, y),

y(t0 ) = y0 ,

tr♦♥❣ ✤â f, g ❧✐➯♥ tư❝ ❝ị♥❣ ✈ỵ✐ ❝→❝ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝➜♣ ♠ët ❝õ❛ ❝❤ó♥❣ tr♦♥❣
❧➙♥ ❝➟♥ ❝õ❛ ✤✐➸♠ (t0 , x0 ), (t0 , y0 )✱ ❦❤✐ ✤â ❤➺ ✤➣ ❝❤♦ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t tr➯♥

✷✶


ởt ừ trử t ự t0 ỵ tỗ t ữỡ
t
ợ ữợ h ❝❤ó♥❣ t❛ s➩ ①➜♣ ①➾ ❝→❝ ❣✐→ trà x(t), y(t) ð ❝→❝ ✤✐➸♠
t1 , t2 , t3 , ... tr♦♥❣ ✤â tn+1 = tn + h ✈ỵ✐ n ≥ 0✳

✶✳✻✳✶

P❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r

❈ỉ♥❣ t❤ù❝ ❧➦♣ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r ✤è✐ ợ ữỡ tr

xn+1 = xn + hf (tn , xn , yn ),
yn+1 = yn + hg(tn , xn , yn ),

P❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r ❝↔✐ t✐➳♥✳ P❤÷ì♥❣ ♣❤→♣ ♥➔② ①→❝ ✤à♥❤ ♥❣❤✐➺♠ ❣➛♥
✤ó♥❣ t❤ỉ♥❣ q✉❛ ✤↕✐ ❧÷đ♥❣ tr✉♥❣ ❣✐❛♥ ❣å✐ ❧➔ ♣❤➛♥ ❞ü ✤♦→♥ ✭❤❛② t✐➯♥ ✤♦→♥✮✳
❈æ♥❣ tự ừ ữỡ r t ố ợ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐
♣❤➙♥ ❧➔✿
P❤➛♥ ❞ü ✤♦→♥✿
un+1 = xn + hf (tn , xn , yn ),
vn+1 = yn + hg(tn , xn , yn ).

P❤➛♥ ❤✐➺✉ ❝❤➾♥❤✿
h

xn+1 = xn + [f (tn , xn , yn ) + f (tn+1 , un+1 , vn+1 )],
2
h
yn+1 = yn + [g(tn , xn , yn ) + g(tn+1 , un+1 , vn+1 )].
2

✶✳✻✳✷

P❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡✲❑✉tt❛

●✐↔ sû t❛ ✤➣ t➼♥❤ ✤÷đ❝ ❝→❝ ❣✐→ trà ❣➛♥ ✤ó♥❣ x1 , y1 , x2 , y2 , ..., xn , yn ❝õ❛
❝→❝ ❣✐→ trà ❝❤➼♥❤ ①→❝ x(t1 ), y(t1 ), x(t2 ), y(t2 ), ..., x(tn ), y(tn ) ✈➔ t❛ ❝➛♥ t➼♥❤

✷✷


t✐➳♣ xn+1 ≈ x(tn+1 ), yn+1 ≈ y(tn+1 )✳ ❑❤✐ õ t ỵ ỡ ừ
t

tn+1

x(tn+1 ) x(tn ) =

tn +h

x (t)dt =
tn

x (t)dt
tn

⑩♣ ❞ö♥❣ q✉② t➢❝ ❙✐♠♣s♦♥ t❛ ❝â✿
x(tn+1 ) − x(tn ) ≈

h
h
[x (tn ) + 4x (tn + ) + x (tn+1 )].
6
2

❙✉② r❛✿
h
h
h
xn+1 ≈ xn + [x (tn ) + 2x (tn + ) + 2x (tn + ) + x (tn+1 )].
6
2
2

❚❤➳ ❝→❝ ✤↕✐ ❧÷đ♥❣ x (tn ), x (tn + h2 ), x (tn + h2 ), x (tn+1 ) ð ❜✐➸✉ t❤ù❝ tr➯♥ ❧➛♥
❧÷đt ❜ð✐ F1 , F2 , F3 , F4 ✤÷đ❝ ①→❝ ✤à♥❤ t❛ ✤÷đ❝ ❝ỉ♥❣ t❤ù❝ ❧➦♣ ❝õ❛ ♣❤÷ì♥❣
♣❤→♣ ❘✉♥❣❡✲❑✉tt❛✳
❈ỉ♥❣ t❤ù❝ ❧➦♣ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡✲❑✉tt❛ ✤è✐ ợ ữỡ tr

h
xn+1 = xn + (F1 + 2F2 + 2F3 + F4 ),
6
h
yn+1 = yn + (G1 + 2G2 + 2G3 + G4 ),
6

tr♦♥❣ ✤â✱
F1 = f (tn , xn , yn ),
h
F2 = f (tn + , xn +
2
h
F3 = f (tn + , xn +
2

h
F1 , yn +
2
h
F2 , yn +
2

h
G1 ),
2
h
G2 ),
2

F4 = f (tn + h, xn + hF3 , yn + hG3 ),
G1 , G2 , G3 , G4 ❧➔ ❝→❝ ❣✐→ trà ❝õ❛ ❤➔♠ g ✤÷đ❝ ①→❝ ✤à♥❤ t÷ì♥❣ tü✳

✷✸

❈❤÷ì♥❣ ✷
Ù♥❣ ❞ư♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛
❝❤♦ ♠ët sè ✤➦❝ tr÷♥❣ ừ ởt ợ
ữỡ tr t t →
t✉②➳♥ t➼♥❤
❚r♦♥❣ ♣❤➛♥ ♥➔② t❛ sû ❞ư♥❣ tr÷í♥❣ ✈➨❝ tì ✤÷đ❝ ✈➩ tr♦♥❣ ▼❛t❤❡♠❛t✐❝❛ ✤➸
♥❣❤✐➯♥ ❝ù✉ ♠ët sè ✤➦❝ tr÷♥❣ ừ ữỡ tr ợ
rsPt
Pttr ❣✭①✱②✮⑥✱ ④①✱ ✲✸✱ ✸⑥✱ ④②✱ ✲✸✱ ✸⑥✱ Pr♦❧♦❣ ✲❃ ④❚❤✐❝❦✲
♥❡ss❬✵✳✵✵✶❪⑥✱ ❆s♣❡❝t❘❛t✐♦ ✲❃✶✱ P❧♦tP♦✐♥ts ✲❃ ✷✵✱ ❋r❛♠❡ ✲❃ ❚r✉❡✱ ❆①❡s ✲❃
❚r✉❡✱ ❆①❡s▲❛❜❡❧ ✲❃ ④①✱ ②⑥❪

✷✹


✷✳✶ ❱➲ ♠ët sè ✤➦❝ tr÷♥❣ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤
t t


ử ỵ ❧♦↕✐ ✤✐➸♠ tỵ✐ ❤↕♥ (0, 0) ✈➔
♥â✐ rã ♥â ❧➔ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥✱ ê♥ ✤à♥❤ ❤❛② ❦❤æ♥❣ ê♥ ✤à♥❤✳
❑✐➸♠ tr ỗ t

ử t



dx
dt

= 2x + y,

dy
dt

✭✷✳✶✮

= x − 2y.

P❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❝õ❛ ❤➺ ✭✷✳✶✮✿
−2 − λ

1

1

−2 − λ

= 0 ⇔ (−2 − λ)2 − 1 = 0.

●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t❛ ♥❤➟♥ ✤÷đ❝ λ1 = −1 ✈➔ λ2 = −3✳ ❙✉② r❛ ✤✐➸♠ tỵ✐ ❤↕♥
(0, 0) ❝õ❛ ❤➺ ✭✷✳✶✮ ❧➔ ♥ót ♣❤✐ ❝❤➼♥❤✱ ❧➔ ✤➾♥❤ ❝❤➻♠ ✈➔ t✐➺♠ ❝➟♥ ê♥ ✤à♥❤✳

❑✐➸♠ tr❛ ❧↕✐ ❜➡♥❣ ỗ t

rữớ tỡ ừ

ừ

ã ử ữỡ ❝❤➼♥❤ ①→❝✿

❚❛ ❝â✿

dx
dt

= −2x + y ⇒

d2 x
dt2

= −2 dx
dt +

✷✺

dy
dt .