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◆ ữợ ❚❙✳ ▲➯ ❍↔✐ ❚r✉♥❣ ❙✐♥❤ ✈✐➯♥ t❤ü❝ ❤✐➺♥ ✿ ◆❣✉②➵♥ ỗ ợ ử õ ỵ ❞♦ ❝❤å♥ ✤➲ 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 tợ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✺✳✶ ❚✉②➳♥ t➼♥❤ ❤â❛ t↕✐ ✤✐➸♠ tỵ✐ ❤↕♥ ✳ ✳ ✳ ✳ ✳ ỵ ✶ ✭❙ü ê♥ ✤à♥❤ ❝õ❛ ❤➺ t✉②➳♥ t➼♥❤✮ ✳ ✳ ỵ ỹ ê♥ ✤à♥❤ ❝õ❛ ❤➺ → t✉②➳♥ t➼♥❤✮ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✻ ▼ët ✈➔✐ ♣❤÷ì♥❣ ♣❤→♣ sè ❣✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✳ ✳ ✳ ✷✶ ✶✳✻✳✶ P❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✷✷ ✶✳✻✳✷ P❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡✲❑✉tt❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷ Ù♥❣ ❞ö♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ❝❤♦ ♠ët số trữ ởt ợ ữỡ tr ♣❤➙♥ t✉②➳♥ t➼♥❤ ✈➔ → t✉②➳♥ t➼♥❤ ✷✳✶ ✷✹ ❱➲ ♠ët sè ✤➦❝ tr÷♥❣ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t t ỵ ①→❝ ✤à♥❤ ❧♦↕✐ ✤✐➸♠ tỵ✐ ❤↕♥ (0, 0) ✈➔ ♥â✐ rã ♥â ❧➔ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥✱ ê♥ ✤à♥❤ ❤❛② ổ tr ỗ t ✳ ✳ ✳ ✳ ✳ ✷✳✶✳✷ ✷✺ ⑩♣ ❞ö♥❣ ✤à♥❤ ỵ ợ tợ (x0 , y0 ) ✤➸ ♣❤➙♥ ❧♦↕✐ ✈➔ ①➨t t➼♥❤ ê♥ ✤à♥❤✳ 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❛ ♣❤÷ì♥❣ ✈➔ tợ t ý ữủ ①→❝ ✤à♥❤ tr♦♥❣ ↔♥❤ ♣❤❛✳ ✳ ✷✳✷✳✷ ✸✼ ❳→❝ ✤à♥❤ tợ ữ r ự ❧♦↕✐ ✈➔ t➼♥❤ ê♥ ✤à♥❤ ❝õ❛ tø♥❣ ✤✐➸♠✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ❙ü ♣❤➙♥ ♥❤→♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✷✳✸ Ù♥❣ ❞ö♥❣✿ ❙ü sè♥❣ sât ❝õ❛ ♠ët ❧♦➔✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✷✳✷✳✸ ❑➳t ❧✉➟♥ ✺✵ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✺✶ P❤ư ❧ư❝ ✺✷ ✸ ▲í✐ ♥â✐ ỵ t 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 t t tr t ỵ ✈➔ s✐♥❤ ❤å❝✳ ❚❤æ♥❣ q✉❛ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ t➼♥❤ ❝❤➜t ❝õ❛ ❤➺ ♥➔② t❛ ❝â t❤➸ ♠✐➯✉ t↔ ♠ët ❝→❝❤ ❝ö t❤➸ ❝→❝ t➼♥❤ ❝❤➜t✱ ✤➦❝ ✤✐➸♠ ❝õ❛ ✈➜♥ ✤➲ ✈➔ ❣✐↔✐ q✉②➳t ✤÷đ❝ ✈➜♥ ✤➲ ✤÷đ❝ ✤➲ ❝➟♣✳ ✣✐➸♠ tợ ữỡ tr t t ✈➔ → t✉②➳♥ t➼♥❤ ❝â ✈à tr➼ q✉❛♥ trå♥❣ ✈➔ ✤â♥❣ ✈❛✐ trá t❤❡♥ ❝❤èt tr♦♥❣ ✈✐➺❝ ♠æ t↔ ❝→❝ tổ q ỗ t ✤â✳ ❚❤➜② ✤÷đ❝ ✈❛✐ trá q✉❛♥ trå♥❣ ✤â ✈➔ ✤÷đ❝ sỹ ủ ỵ t ữợ ❞➝♥ ✲ ❚❙✳ ▲➯ ❍↔✐ ❚r✉♥❣ ♥➯♥ tæ✐ ❝❤å♥ ✤➲ t tợ ởt ợ ữỡ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ 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 ❝ỉ♥❣ ❝ư ❦❤→ ♠↕♥❤ ✤÷đ❝ ❞ị♥❣ ♥❤✐➲✉ tr♦♥❣ ❦ÿ tt ữ tr t t õ ỏ ợ ♠➫✳ ❚❤ỉ♥❣ q✉❛ ✈✐➺❝ sû ❞ư♥❣ ♣❤➛♥ ♠➲♠ ♥➔② t❤➻ ❝ỉ♥❣ ✈✐➺❝ t➼♥❤ t♦→♥ ❝õ❛ ❝❤ó♥❣ t❛ s➩ ♥❤❛♥❤ ❝❤â♥❣ ✈➔ ✤ì♥ ❣✐↔♥ ❤ì♥✳ ✵✳✷ ▼ư❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ ❚r➯♥ ỡ s ự ỵ sỹ t t ự ỵ sü ê♥ ✤à♥❤ ❝õ❛ ❤➺ → t✉②➳♥ t➼♥❤ t❤æ♥❣ q✉❛ ❝→❝ ✈➼ ❞ö ❝ö 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 ❝→❝ ✈➼ ❞ư ❝ư t❤➸✳ ✵✳✹ ✣è✐ t÷đ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ ◆❣❤✐➯♥ ❝ù✉ ✤✐➸♠ tỵ✐ ❤↕♥ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ✈➔ → t✉②➳♥ t➼♥❤ ❤❛✐ ❜✐➳♥ t❤ỉ♥❣ q✉❛ ❝→❝ ✈➼ ❞ư ❝ư t❤➸✳ ❳➙② ❞ü♥❣ ❝→❝ ❝➙✉ ❧➺♥❤ tr♦♥❣ ♠❛t❤❡♠❛t✐❝❛ ✺✳✷ ✤➸ ✈➩ tr÷í♥❣ ✈➨❝ tì ✈➔ ❣✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❜➡♥❣ ♠ët ✈➔✐ ♣❤÷ì♥❣ ♣❤→♣ sè✳ ✺ ✵✳✺ ❈➜✉ tró❝ ❧✉➟♥ ✈➠♥ ◆❣♦➔✐ ♣❤➛♥ ♠ð ✤➛✉✱ ❦➳t ❧✉➟♥✱ t➔✐ ❧✐➺✉ t❤❛♠ ử tr ỗ õ ❝→❝ ❝❤÷ì♥❣ s❛✉✿ ❈❤÷ì♥❣ ✶✿ ▼ët sè ❦❤→✐ ♥✐➺♠ ♠ð ✤➛✉✳ ❈❤÷ì♥❣ ✷✿ Ù♥❣ ❞ư♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ❝❤♦ ♠ët sè ✤➦❝ tr÷♥❣ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ✈➔ → t✉②➳♥ t➼♥❤✳ ▲❮■ ❈❷▼ ❒◆ ❚→❝ ❣✐↔ ①✐♥ t❤➸ ❤✐➺♥ ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ t❤➛② ữợ r õ ỳ ủ ỵ õ õ qỵ tr q tr tỹ t ỗ tớ t ụ ❣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 n2 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➯♥ ✤÷đ❝ ❣✐↔ t❤✐➳t ❧✐➯♥ tư❝ tr➯♥ ❦❤♦↔♥❣ I = (a; b) R ũ ỵ tr t õ t t ữợ y = A(x)y + g(x), tr♦♥❣ ✤â 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 tợ ổ ❧➟♣ ♥➳✉ t❤ä❛  g(x , y ) = 0 t tợ ã út tợ (x0 , y0 ) ữủ ❧➔ ✤✐➸♠ ♥ót ♥➳✉ t❤ä❛ ♠➣♥ ✷ ✤✐➲✉ ❦✐➺♥✿ ✕ ▼å✐ q✉ÿ ✤↕♦ ✤➲✉ t✐➳♥ tỵ✐ (x0 , y0 ) ❦❤✐ t → +∞ ❤♦➦❝ ♠å✐ q✉ÿ ✤↕♦ ✤➲✉ rí✐ ①❛ (x0 , y0 ) ❦❤✐ t → +∞✳ ✕ q t ú ợ ữớ 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❤➻ õ ữủ út ỗ ã s ◆➳✉ ❝ù ♠é✐ ❝➦♣ q✉ÿ ✤↕♦ ✤è✐ ❞✐➺♥ ❦❤→❝ ♥❤❛✉ ổ õ t ú ợ ữớ t q tợ t út ữủ ❧➔ ✤✐➸♠ ❤➻♥❤ s❛♦ ❤❛② ✤✐➸♠ s❛♦✳ ✾ • ◆ót ♣❤✐ ❝❤➼♥❤✿ ▼å✐ q✉ÿ ✤↕♦ trø r❛ ♠ët ❝➦♣ q✉ÿ ố t ú ợ ởt ữớ t ✤✐ q✉❛ ✤✐➸♠ tỵ✐ ❤↕♥ t❤➻ ♥ót ✤â ❣å✐ ❧➔ út ã ỹ tợ ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ ②➯♥ ♥❣ü❛ ♥➳✉ ♠å✐ q✉ÿ ✤↕♦ ❧➔ ❜→♥ trö❝ ❝õ❛ ❤②♣❡❜♦❧✱ ✤✐➸♠ (x(t), y(t)) ❞➛♥ ✤➳♥ (x0 , y0 ) t❤❡♦ trư❝ Ox ✈➔ rí✐ ①❛ t❤❡♦ trö❝ Oy ❦❤✐ t → +∞ ✈➔ ❝â ❤❛✐ q✉ÿ ✤↕♦ ❞➛♥ ✤➳♥ (x0 , y0 ) ♥❤÷♥❣ ✤➲✉ ổ t + ã tợ tợ ữủ ✤à♥❤ ♥➳✉ ❦❤✐ ✤✐➸♠ ✤➛✉ (x1 , y1 ) ✤õ ❣➛♥ (x0 , y0 ) t❤➻ ✤✐➸♠ (x(t), y(t)) ❧✉æ♥ ❣➛♥ (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 > : |X0 − X1 | < δ t❤➻ |X0 − X(t)| < ε, ∀t > 0✳ ✣✐➸♠ tỵ✐ ❤↕♥ ❦❤æ♥❣ ê♥ ✤à♥❤ ♥➳✉ ♥â ❦❤æ♥❣ ❧➔ ✤✐➸♠ ê♥ ✤à♥❤ tù❝ ❧➔∃ε > 0, ∃δ > : |X0 − X1 | < δ ♠➔ |X0 − X(t)| < ε, ∀t > 0✳ ◆ót ❧ã♠ ❝â |X0 − X(t)| < ε, ∀t > ❞♦ ✤â ♥â ❧➔ ✤✐➸♠ út ã tợ ữủ ❧➔ t➙♠ ♥➳✉ ♥â ❧➔ ✤✐➸♠ tỵ✐ ❤↕♥ ê♥ ✤à♥❤ ✤÷đ❝ ❜❛♦ q✉❛♥❤ ❜ð✐ ❝→❝ q✉ÿ ✤↕♦ ❦➼♥✱ t✉➛♥ ❤♦➔♥✳ ❙✉② r❛ ♠å✐ t➙♠ ✤➲✉ ê♥ ✤à♥❤✳ • ❚✐➺♠ ❝➟♥ ê♥ ✤à♥❤ ✭ê♥ ✤à♥❤ t✐➺♠ ❝➟♥✮✿ ✣✐➸♠ tỵ✐ ❤↕♥ (x0 , y0 ) ✤÷đ❝ ❣å✐ ❧➔ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ ♥➳✉ ❧➔ ✤✐➸♠ ê♥ ✤à♥❤ ✈➔ ♠å✐ q✉ÿ ✤↕♦ ✤õ ❣➛♥ (x0 , y0 ) ✤➲✉ t✐➳♥ tỵ✐ (x0 , y0 ) ❦❤✐ t → +∞✳ ❍❛② ∃δ > : |X0 − X1 | < δ, lim X(t) = X0 ✳ t→+∞ ✶✵ ❍➻♥❤ ✷✳✾✿ ❚r÷í♥❣ ✈➨❝ tì ❝õ❛ ❤➺ ✭✷✳✶✵✮ ❍➻♥❤ ✷✳✶✵✿ ❚r÷í♥❣ ✈➨❝ tì ❝õ❛ ❤➺ ✭✷✳✾✮ ❍➻♥❤ ✷✳✶✶✿ ❷♥❤ ♣❤❛ ❝õ❛ ❤➺ ✭✷✳✾✮ ◗✉❛ ✤➙② ❝❤♦ t❤➜② ♥❣♦➔✐ ✤✐➸♠ tỵ✐ ❤↕♥ (0; 0) ❧➔ ♥ót ♣❤✐ ❝❤➼♥❤ ê♥ ✤à♥❤ r❛ ❤➺ ✭✷✳✾✮ ❝á♥ ❝â ♠ët ✤✐➸♠ tỵ✐ ❤↕♥ ❣➛♥ ✤✐➸♠ (0, 67; 0, 40) ❧➔ ♠ët ✤✐➸♠ ②➯♥ ♥❣ü❛ ê♥ ✤à♥❤✳ ✷✳✷✳✷ ❳→❝ ✤à♥❤ ❝→❝ ✤✐➸♠ tợ ữ r ự t➼♥❤ ê♥ ✤à♥❤ ❝õ❛ tø♥❣ ✤✐➸♠✳ ❱➼ ❞ö ✷✳✷✳✷✳✶✳ ❳➨t ❤➺✿   dx dt  dy dt = x − y, = x − y ✸✽ ✭✷✳✶✶✮ ❚❛ ❝â✿    x − y = 0,  x2 − x = 0, (1) ⇔  x2 − y =  x − y = (2) ●✐↔✐ ✭✶✮ ✤÷đ❝✿ x = 0✱ x = 1✳ ❚❤➳ x ✈➔♦ ✭✷✮ ✤÷đ❝ y = 0✱ y = 1✳ ❱➟② ❤➺ ✭✷✳✶✶✮ ❝â ✷ ✤✐➸♠ tỵ✐ ❤↕♥ (0, 0) (1, 1) ã ố ợ tợ (0, 0)✿ ❍➺ t✉②➳♥ t➼♥❤ t÷ì♥❣ ù♥❣ ❝õ❛ ❤➺ ✭✷✳✶✶✮ ❧➔✿   dx dt = x − y, dy dt  ✭✷✳✶✷✮ = −y P❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❝õ❛ ❤➺ ✭✷✳✶✷✮ ❧➔✿ 1−λ −1 −1 − λ = ⇔ (1 − λ)(−1 − λ) = ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t❛ ♥❤➟♥ ✤÷đ❝ λ1 = ✈➔ λ2 = −1 ❧➔ ❤❛✐ ♥❣❤✐➺♠ t❤ü❝ tr→✐ ❞➜✉✳ ❙✉② r❛ (0, 0) ❧➔ ✤✐➸♠ ②➯♥ ♥❣ü❛ ❦❤ỉ♥❣ ê♥ ✤à♥❤✳ • ✣è✐ ✈ỵ✐ ✤✐➸♠ tỵ✐ ❤↕♥ (1, 1)✿ ❚❛ ❝â✿ f (x, y) = x − y ✱ g(x, y) = x2 − y ✱ s✉② r❛✿  J(x, y) =  ❉♦ ✤â✿ −1 2x −1  J(1, 1) =  −1 −1     ❍➺ t✉②➳♥ t➼♥❤ t÷ì♥❣ ù♥❣ ❝õ❛ ❤➺ ✭✷✳✶✶✮ ❧➔✿    du dt dv dt = u − v, = 2u − v ✸✾ ✭✷✳✶✸✮ P❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❝õ❛ ❤➺ ✭✷✳✶✸✮ ❧➔✿ 1−λ −1 −1 − λ = ⇔ (1 − λ)(−1 − λ) + = ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t❛ ♥❤➟♥ ✤÷đ❝ λ1 = i ✈➔ λ2 = −i ❧➔ ❤❛✐ ♥❣❤✐➺♠ t❤✉➛♥ ↔♦✳ ❍➻♥❤ ✷✳✶✷✿ ❚r÷í♥❣ ✈➨❝ tì ❝õ❛ ❤➺ ✭✷✳✶✷✮ ❍➻♥❤ ✷✳✶✸✿ ❚r÷í♥❣ ✈➨❝ tì ❝õ❛ ❤➺ ✭✷✳✶✸✮ ❍➻♥❤ ✷✳✶✹✿ ❚r÷í♥❣ ✈➨❝ tì ❝õ❛ ❤➺ ✭✷✳✶✶✮ ❍➻♥❤ ✷✳✶✺✿ ❷♥❤ ♣❤❛ ❝õ❛ ❤➺ ✭✷✳✶✶✮ ❙✉② r❛ ❝→❝ ✤✐➸♠ tỵ✐ ❤↕♥ ❧➔ (0, 0) ❧➔ ✤✐➸♠ ②➯♥ ♥❣ü❛ ❦❤æ♥❣ ê♥ ✤à♥❤ ✈➔ (1, 1) ❧➔ t➙♠✳ ❱➼ ❞ö ✷✳✷✳✷✳✷✳ ❳➨t ❤➺✿    dx dt dy dt = y − 1, = x − y ✹✵ ✭✷✳✶✹✮ ❚❛ ❝â✿     y − = 0,  y = 1,  y = 1, ⇔ ;  x2 − y =  x =  x = −1 ❱➟② ❤➺ ✭✷✳✶✹✮ ❝â ✷ ✤✐➸♠ tỵ✐ ❤↕♥ (1, 1) ✈➔ (−1, 1)✳ ❚❛ ❝â✿ f (x, y) = y − 1✱ g(x, y) = x2 − y ✱ s✉② r❛✿  J(x, y) =    2x ã ố ợ tợ (1, 1) ❝â✿  J(1, 1) =   −1  ❍➺ t✉②➳♥ t➼♥❤ t÷ì♥❣ ù♥❣ ❝õ❛ ❤➺ ✭✷✳✶✹✮ ❧➔✿    du dt dv dt = v, ✭✷✳✶✺✮ = 2u − v P❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❝õ❛ ❤➺ ✭✷✳✶✺✮ ❧➔✿ −λ −1 − λ = ⇔ −λ(−1 − λ) − = ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t❛ ♥❤➟♥ ✤÷đ❝ λ1 = ✈➔ λ2 = −2 ❧➔ ❤❛✐ ♥❣❤✐➺♠ t❤ü❝ tr→✐ ❞➜✉✳ ❙✉② r❛ (1, 1) ❧➔ ✤✐➸♠ ②➯♥ ♥❣ü❛ ❦❤ỉ♥❣ ê♥ ✤à♥❤✳ • ✣è✐ ✈ỵ✐ ✤✐➸♠ tỵ✐ ❤↕♥ (−1, 1)✿ ❚❛ ❝â✿  J(−1, 1) =  ✹✶ −2 −1   ❍➺ t✉②➳♥ t➼♥❤ t÷ì♥❣ ù♥❣ ❝õ❛ ❤➺ ✭✷✳✶✹✮ ❧➔✿    du dt dv dt = v, ✭✷✳✶✻✮ = −2u − v P❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❝õ❛ ❤➺ ✭✷✳✶✻✮ ❧➔✿ −λ = ⇔ −λ(−1 − λ) + = −2 −1 − λ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t❛ ♥❤➟♥ ✤÷đ❝ λ1 = − 21 √ − i ✈➔ λ2 = − 12 √ + i ❧➔ ❤❛✐ ♥❣❤✐➺♠ ♣❤ù❝ ❧✐➯♥ ❤ñ♣ ❝â ♣❤➛♥ t❤ü❝ ➙♠✳ ❙✉② r❛ ✤✐➸♠ tỵ✐ ❤↕♥ (−1, 1) ❧➔ ✤✐➸♠ ①♦➢♥ è❝ ✈➔ t✐➺♠ ❝➟♥ ê♥ ✤à♥❤✳ ❍➻♥❤ ✷✳✶✻✿ ❚r÷í♥❣ ✈➨❝ tì ❝õ❛ ❤➺ ✭✷✳✶✺✮ ❍➻♥❤ ✷✳✶✼✿ ❚r÷í♥❣ ✈➨❝ tì ❝õ❛ ❤➺ ✭✷✳✶✻✮ ✹✷ ❍➻♥❤ ✷✳✶✽✿ ❚r÷í♥❣ ✈➨❝ tì ❝õ❛ ❤➺ ✭✷✳✶✹✮ ✷✳✷✳✸ ❍➻♥❤ ✷✳✶✾✿ ❷♥❤ ♣❤❛ ❝õ❛ ❤➺ ✭✷✳✶✹✮ ❙ü ♣❤➙♥ ♥❤→♥❤ ❚❛ ✤✐ ①➨t ❝→❝ tr÷í♥❣ ❤đ♣ ✤✐➸♥ ❤➻♥❤ ❦❤✐ ❝â sü ①→♦ trë♥ ♥❤ä ❝→❝ ❤➺ sè ❝õ❛ ❤➺ t✉②➳♥ t➼♥❤ ❤♦➦❝ → t✉②➳♥ t➼♥❤ ❝â t❤➸ ❧➔♠ t❤❛② ✤ê✐ t➼♥❤ ê♥ ✤à♥❤ ❤♦➦❝ ❧♦↕✐ ✤✐➸♠ tỵ✐ ❤↕♥✳ ❳➨t ❤➺ t✉②➳♥ t➼♥❤✿    dx dt = −x + y, dy dt = x − y ✭✷✳✶✼✮ ❑❤✐ ✤â ✤✐➸♠ tỵ✐ ❤↕♥ (0, 0) ❧➔✿ ❛✳ ♠ët ✤✐➸♠ ①♦➢♥ è❝ ê♥ ✤à♥❤ ♥➳✉ < 0✳ P❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❝õ❛ ❤➺ ✭✷✳✶✼✮ ❧➔✿ −1 − λ = ⇔ (−1 − λ)2 − = −1 − λ √ < ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t❛ ♥❤➟♥ ✤÷đ❝ λ1 = −1 + i − ✈➔ λ2 = √ −1 − i − ❧➔ ❤❛✐ ♥❣❤✐➺♠ ♣❤ù❝ ❧✐➯♥ ❤ñ♣ ❝â ♣❤➛♥ t❤ü❝ ➙♠✳ ❙✉② r❛ ✤✐➸♠ ❱ỵ✐ tỵ✐ ❤↕♥ (0, 0) ❧➔ ✤✐➸♠ ①♦➢♥ è❝ ê♥ ✤à♥❤✳ ❜✳ ♥ót ê♥ ✤à♥❤ ♥➳✉ ≤ < 1✳ ◆❣♦➔✐ r❛ sü ①→♦ trë♥ ♥❤ä ❝õ❛ ❤➺ x = −x, y = x − y ❝â t❤➸ ❧➔♠ t❤❛② ✤ê✐ ❦✐➲✉ ✤✐➸♠ tỵ✐ ❤↕♥ (0, 0) ♠➔ ❦❤ỉ♥❣ ✹✸ ❧➔♠ ♠➜t t➼♥❤ ê♥ ✤à♥❤ ❝õ❛ ♥â✳ ❱ỵ✐ ≤ λ2 = −1 − < ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t❛ ♥❤➟♥ ✤÷đ❝ λ1 = −1 + √ √ ✈➔ ❧➔ ❤❛✐ ♥❣❤✐➺♠ t❤ü❝ ✤➲✉ ➙♠✳ ❙✉② r❛ ✤✐➸♠ tỵ✐ ❤↕♥ (0, 0) ❧➔ ♥ót ê♥ ✤à♥❤✳ ◆➳✉ < < t❤➻ ✤✐➸♠ tỵ✐ ❤↕♥ (0, 0) ❧➔ ♥ót ♣❤✐ ❝❤➼♥❤ ê♥ ✤à♥❤✳ ◆➳✉ = t❛ ✤÷đ❝ ❤➺ x = −x, y = x − y ❝â ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❝â ♥❣❤✐➺♠ t❤ü❝ ➙♠ λ = −1 ➙♠✱ ❞♦ ✤â ✤✐➸♠ tỵ✐ ❤↕♥ (0, 0) ❧➔ ê♥ ✤à♥❤ ♥❤÷♥❣ ❦✐➸✉ ✤✐➸♠ t❤❛② ✤ê✐ ❧➔ ♥ót ❝❤➼♥❤ ❤♦➦❝ ♣❤✐ ❝❤➼♥❤✳ ❍➻♥❤ ✷✳✷✵✿ ❚r÷í♥❣ ✈➨❝ tì ❝õ❛ ❤➺ ợ = 0, 05 rữớ tỡ ợ = 0, rữớ ✈➨❝ tì ❝õ❛ ❤➺ ✭✷✳✶✼✮ ✈ỵ✐ = ❍➻♥❤ ✷✳✷✸✿ rữớ tỡ ợ = 0, 05 rữớ tỡ ợ = 0, ✷✳✸ Ù♥❣ ❞ö♥❣✿ ❙ü sè♥❣ sât ❝õ❛ ♠ët ❧♦➔✐ ❳➨t ❤❛✐ ❧♦➔✐ ✭✈➼ ❞ö ✤ë♥❣ ✈➟t✱ ❝➙② ❝è✐ ❤♦➦❝ ✈✐ ❦❤✉➞♥✮ ✈ỵ✐ tê♥❣ t❤➸ x(t) ✈➔ y(t) t↕✐ t❤í✐ ❣✐❛♥ t✱ ❝↕♥❤ tr❛♥❤ ✈ỵ✐ ♥❤❛✉ ✤➸ t➻♠ t❤ù❝ ➠♥ ❝â s➤♥ tr♦♥❣ ♠ỉ✐ tr÷í♥❣ ❝❤✉♥❣ ❝õ❛ ❝❤ó♥❣✳ ✣➸ ①➙② ❞ü♥❣ ♠ët ♠æ ❤➻♥❤ t♦→♥ ❤➳t sù❝ t❤ü❝ t➳✱ t❛ ❣✐↔ sû sü ✈➢♥❣ ♠➦t ❝õ❛ ♠ët ❧♦➔✐ ♥➔♦ ✤â t❤➻ ❧♦➔✐ ❦❤→❝ ❝â tê♥❣ t❤➸ t➠♥❣ ❧➯♥✳ ◆➳✉ ❦❤ỉ♥❣ ❝â sü t÷ì♥❣ t→❝ ❣✐ú❛ ❤❛✐ ❧♦➔✐✱ tê♥❣ t❤➸ ❝õ❛ ❝❤ó♥❣ x(t), y(t) t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✿   dx dt  dy dt = a1 x − b1 x2 , = a2 y − b2 y ✭✷✳✶✽✮ ◆➳✉ ❝✉ë❝ ❝↕♥❤ tr❛♥❤ ↔♥❤ ❤÷ð♥❣ ✤➳♥ tè❝ ✤ë s✉② t❤♦→✐ tr♦♥❣ ♠é✐ tê♥❣ t❤➸ ✈➔ t ợ t xy ữủ ❝↕♥❤ tr❛♥❤✿   dx dt  dy dt = a1 x − b1 x2 − c1 xy, = a2 y − b2 y − c2 xy, ✭✷✳✶✾✮ tr♦♥❣ ✤â ❝→❝ ❤➺ sè a1 , a2 , b1 , b2 , c1 , c2 ✤➲✉ ❞÷ì♥❣✳ ●✐↔ sû t❛ ❝â ♠ët tê♥❣ t❤➸ x(t) ✈➔ y(t) t❤ä❛ ♠➣♥ ❤➺✿   dx dt  dy dt = 14x − 21 x2 − xy, = 16y − ✹✺ 2y − xy ✭✷✳✷✵✮ ●✐↔✐ ❤➺✿   14x − x2 − xy = 0,  16y − y − xy = 0, ✤÷đ❝ ❝→❝ ♥❣❤✐➺♠ (0, 0), (0, 32), (28, 0), (12, 8)✳ ❱➟② ❤➺ ✭✷✳✷✵✮ ❝â ✹ ✤✐➸♠ tỵ✐ ❤↕♥ (0, 0), (0, 32), (28, 0), (12, 8) ã ố ợ tợ ❤↕♥ (0, 0)✿ ❍➺ t✉②➳♥ t➼♥❤ t÷ì♥❣ ù♥❣ ❝õ❛ ❤➺ ✭✷✳✷✵✮ ❧➔✿   dx dt = 14x,  dy dt = 16y ✭✷✳✷✶✮ P❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❝õ❛ ❤➺ ✭✷✳✷✶✮ ❧➔✿ 14 − λ 0 16 − λ = ⇔ (14 − λ)(16 − λ) = ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t❛ ♥❤➟♥ ✤÷đ❝ λ1 = 14 ✈➔ λ2 = 16 ❧➔ ❤❛✐ ♥❣❤✐➺♠ t❤ü❝ ✤➲✉ ❞÷ì♥❣✳ ❙✉② r (0, 0) út ỗ ổ ã ✣è✐ ✈ỵ✐ ✤✐➸♠ tỵ✐ ❤↕♥ (0, 32)✿ ❚❛ ❝â✿ f (x, y) = 14x − 12 x2 − xy, g(x, y) = 16y − 12 y − xy ✱ s✉② r❛✿  J(x, y) =  14 − x − y −x −y 16 − y − x   ❉♦ ✤â✿ J(0, 32) =  −18    −32 −16 ❍➺ t✉②➳♥ t➼♥❤ t÷ì♥❣ ù♥❣ ❝õ❛ ❤➺ ✭✷✳✷✵✮ ❧➔✿    du dt dv dt = −18u, = −32u − 16v ✹✻ ✭✷✳✷✷✮ P❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❝õ❛ ❤➺ ✭✷✳✷✷✮ ❧➔✿ −18 − λ −32 −16 − λ = ⇔ (−18 − λ)(−16 − λ) = ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t❛ ♥❤➟♥ ✤÷đ❝ λ1 = −18 ✈➔ λ2 = −16 ❧➔ ❤❛✐ ♥❣❤✐➺♠ t❤ü❝ ➙♠✳ ❙✉② r❛ ✤✐➸♠ tỵ✐ ❤↕♥ (0, 32) ❧➔ út ã ố ợ tợ (28; 0)✿ ❚❛ ❝â✿   −14 −28  J(28, 0) =  −12 ❍➺ t✉②➳♥ t➼♥❤ t÷ì♥❣ ù♥❣ ❝õ❛ ❤➺ ✭✷✳✷✵✮ ❧➔✿   du dt  = −14u − 28v, dv dt = −12v ✭✷✳✷✸✮ P❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❝õ❛ ❤➺ ✭✷✳✷✸✮ ❧➔✿ −14 − λ −28 −12 − λ = ⇔ (−14 − λ)(−12 − λ) = ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t❛ ♥❤➟♥ ✤÷đ❝ λ1 = −14 ✈➔ λ2 = −12 ❧➔ ❤❛✐ ♥❣❤✐➺♠ t❤ü❝ ➙♠✳ ❙✉② r❛ ✤✐➸♠ tỵ✐ ❤↕♥ (28, 0) ❧➔ ♥ót ã ố ợ tợ (12, 8) ❚❛ ❝â✿  J(12, 8) =  ✹✼ −6 −12 −8 −4   ❍➺ t✉②➳♥ t➼♥❤ t÷ì♥❣ ù♥❣ ❝õ❛ ❤➺ ✭✷✳✷✵✮ ❧➔✿   du dt  dv dt = −6u − 12v, = −8u − 4v ✭✷✳✷✹✮ P❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❝õ❛ ❤➺ ✭✷✳✷✹✮ ❧➔✿ −6 − λ −12 −8 −4 − λ = ⇔ (−6 − λ)(−4 − λ) − 96 = √ √ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t❛ ♥❤➟♥ ✤÷đ❝ λ1 = −5 − 97 ✈➔ λ2 = −5 + 97 ❧➔ ❤❛✐ ♥❣❤✐➺♠ t❤ü❝ tr→✐ ❞➜✉✳ ❙✉② r❛ ✤✐➸♠ tỵ✐ ❤↕♥ (12, 8) ❧➔ ✤✐➸♠ ②➯♥ ♥❣ü❛ ❦❤æ♥❣ ê♥ ✤à♥❤✳ ❍➻♥❤ ✷✳✷✺✿ ❷♥❤ ♣❤❛ ❝õ❛ ❤➺ ✭✷✳✷✵✮ ◗✉❛ ↔♥❤ ♣❤❛ ❝õ❛ ❤➺ ✭✷✳✷✵✮ t❤➻ ❝â ❤❛✐ q✉ÿ ✤↕♦ ❞➛♥ ✤➳♥ ✤✐➸♠ ②➯♥ ♥❣ü❛ (12, 8) ũ ợ ỹ t ữớ t→❝❤ t❤➔♥❤ ❤❛✐ ♠✐➲♥ ✤â♥❣ ✈❛✐ trá q✉❛♥ trå♥❣ ✤➸ ①→❝ ✤à♥❤ ❤➔♥❤ ✤ë♥❣ ❧➙✉ ❞➔✐ ❝õ❛ tê♥❣ t❤➸✳ • ◆➳✉ ✤✐➸♠ ✤➛✉ (x0 , y0 ) ♥➡♠ tr➯♥ ✤÷í♥❣ t→❝❤ t❤➻ (x(t), y(t)) ❞➛♥ ✤➳♥ (12, 8) ❦❤✐ t → +∞✳ ❉➽ ♥❤✐➯♥ ❝→❝ sü ✈✐➺❝ ♥❣➝✉ ♥❤✐➯♥ ❧➔♠ õ ổ ố ợ (x(t), y(t)) tr ữớ t ổ sỹ ũ tỗ t ỏ ❧➔ ❦❤ỉ♥❣ t❤➸✳ ✹✽ • ◆➳✉ (x0 , y0 ) ♥➡♠ tr♦♥❣ ♠✐➲♥ ■ tr➯♥ ✤÷í♥❣ t→❝❤ t❤➻ (x(t), y(t)) ❞➛♥ ✤➳♥ (0, 32) ❦❤✐ t → +∞✳ ❱➻ ✈➟② tê♥❣ t❤➸ x(t) ❣✐↔♠ ✈➲ ✵✳ • ◆➳✉ (x0 , y0 ) tr ữợ ữớ t t (x(t), y(t)) ❞➛♥ ✤➳♥ (28, 0) ❦❤✐ t → +∞✳ ❱➻ ✈➟② tê♥❣ t❤➸ y(t) ❞✐➺t ✈♦♥❣✳ ✹✾ ❑➳t ❧✉➟♥ ữủ t tợ tợ tr ữỡ tr q ✤â ❝❤♦ t❤➜② ✈à tr➼ q✉❛♥ trå♥❣ ❝õ❛ ✤✐➸♠ tỵ✐ ❤↕♥ tr♦♥❣ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ✤➦❝ tr÷♥❣ ❝õ❛ ❤➺ t✉②➳♥ t t t ổ q ỵ sỹ t t ỵ ✷ ✭sü ê♥ ✤à♥❤ ❝õ❛ ❤➺ → t✉②➳♥ t➼♥❤✮ t❛ ❝â t❤➸ ❦➳t ❧✉➟♥ ❝→❝ ✤➦❝ tr÷♥❣ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ♠ët ❝→❝❤ ♥❤❛♥❤ ❝❤â♥❣ ✈➔ ❞➵ ❞➔♥❣✳ ✣✐➲✉ ♥➔② ❝â ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ♠æ ❤➻♥❤ ❜✐➸✉ ❞✐➵♥ ❜ð✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ♥❤÷✿ ▼ỉ ❤➻♥❤ s✐♥❤ t❤→✐ ❝❤✐➳♠ ✤♦↕t✱ ❝↕♥❤ tr❛♥❤✱ ❜ị♥❣ ♥ê ❞➙♥ sè✳✳✳ ❙û ❞ư♥❣ ✤÷đ❝ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ✺✳✷ ✤➸ ①➙② ❞ü♥❣ ↔♥❤ ♣❤❛ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✱ ✈➟♥ ❞ư♥❣ s♦ s→♥❤ ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ sè ✤➸ ❣✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✈ỵ✐ ❜➔✐ t♦→♥ ❣✐→ trà ✤➛✉✳ ▼➦❝ ❞ị ✈➟② ❧✉➟♥ ✈➠♥ ✈➝♥ ❝á♥ ♠ët sè ❤↕♥ ❝❤➳ ❧➔ ❝❤÷❛ ❦❤❛✐ t❤→❝ ✤÷đ❝ ♥❤✐➲✉ ♣❤÷ì♥❣ ♣❤→♣ sè ✈➔ ✈➟♥ ❞ư♥❣ ♣❤➛♥ ♠➲♠ ✤➸ ✈➩ q✉ÿ ✤↕♦ ❝õ❛ ❝→❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ♥❤➡♠ t❤➸ ❤✐➺♥ rã ♥➨t ❤ì♥ t➼♥❤ ❝❤➜t ❝õ❛ ❤➺ ✤➣ ❝❤♦✳ ❑➼♥❤ ♠♦♥❣ sü ✤â♥❣ ❣â♣ þ ❦✐➳♥ ❝õ❛ q✉þ t❤➛② ❝æ ✤➸ ❧✉➟♥ ✈➠♥ ❝õ❛ ❡♠ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ✣➔ ◆➤♥❣✱ ♥❣➔② ✶✽ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✶✷✳ ❙✐♥❤ tỹ ỗ t❤❛♠ ❦❤↔♦ ❬✶❪ ❆r♥♦❧❞ ❱✳■✳ ✭✶✾✼✽✮✱ ❖r❞✐♥❛r② ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✱ ▼■❚✳ ❬✷❪ ❈✳ ❍❡♥r② ❊❞✇❛r❞s✱ ❊✳ P❡♥♥❡② ❉❛✈✐❞ ✭✷✵✵✼✮✱ ❡q✉❛t✐♦♥s ✇✐t❤ ❜♦✉♥❞❛r② ✈❛❧✉❡✱ Pr❡♥t✐❝❡ ▼❛❧❧✳ ❬✸❪ ❲✐❧❧✐❛♠ ❊✳ ❇♦②❝❡ ✭✷✵✵✵✮✱ ❊❧❡♠❡♥t❛r② ❞✐❢❢❡r❡♥t✐❛❧ ❊❧❡♠❡♥t❛r② ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✇✐t❤ ❜♦✉♥❞❛r② ✈❛❧✉❡✱ ❏♦♥❤ ❲✐❧❡② ❛♥❞ ❙♦♥s✳ ❬✹❪ ❇↔♥ ❞à❝❤ ✭✷✵✵✽✮✱ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝ì ❜↔♥ ✈ỵ✐ ❜➔✐ t♦→♥ ❣✐→ trà ❜✐➯♥✲ ❚➟♣ ✷✱ ✣↕✐ ❤å❝ ❚❤õ② ▲đ✐✳ ❬✺❪ ❤tt♣✿✴✴r❡❢❡r❡♥❝❡✳✇♦❧❢r❛♠✳❝♦♠✴♠❛t❤❡♠❛t✐❝❛✴❤♦✇t♦✴P❧♦t❆❱❡❝t♦r❋✐❡❧❞✳❤t♠❧✳ ❬✻❪ ❤tt♣✿✴✴✺✺❝❧❝✷✳✇♦r❞♣r❡ss✳❝♦♠✴✷✵✶✶✴✵✶✴✶✾✴♠ët✲sè✲❧➺♥❤✲❝ì✲❜↔♥✲tr♦♥❣✲ ♠❛t❤❡♠❛t✐❝❛✴✳ ✺✶ P❤ư ỵ ỹ tỗ t ữỡ ❈❤♦ f ❧➔ ♠ët ❤➔♠ ❣✐→ trà ✈➨❝ tì ✭✈ỵ✐ m t❤➔♥❤ ♣❤➛♥✮ ❝õ❛ m + ❝→❝ ❜✐➳♥ sè t❤ü❝ x1 , x2 , , xm ✈➔ t✳ ◆➳✉ r t f tỗ t ✈➔ ❧✐➯♥ tö❝ tr♦♥❣ ♠ët ❧➙♥ ❝➟♥ ❝õ❛ ✤✐➸♠ x = b, t = a✱ t❤➻ ❜➔✐ t♦→♥ ❣✐→ trà ❜❛♥ ✤➛✉✿ dx = f (x, t), dt x(a) = b ❝â ♠ët ♥❣❤✐➺♠ tr➯♥ ♠ët ❦❤♦↔♥❣ ♥➔♦ ✤â ❝❤ù❛ t = a ỵ t ❝õ❛ ♥❣❤✐➺♠✮ ❈❤♦ ❜➔✐ t♦→♥ ❣✐→ trà ❜❛♥ ✤➛✉✿ dx = f (x, t), dt x(a) = b ●✐↔ sû tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ Rm+1 ❤➔♠ f ❧✐➯♥ tư❝ ✈➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ▲✐♣s❝❤✐t③✿ |f (x1 , t) − f (x2 , t)| ≤ k|x1 − x2 | ✈ỵ✐ x1 , x2 ❧➔ ✷ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❣✐→ trà ❜❛♥ ✤➛✉✳ ◆➳✉ x1 (t), x2 (t) ❧➔ ✷ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ tr➯♥ ♠ët ❦❤♦↔♥❣ ♠ð I ❝❤ù❛ x = a✱ s❛♦ ❝❤♦ ❝→❝ ✤÷í♥❣ ❝♦♥❣ ♥❣❤✐➺♠ (x1 (t), t) ✈➔ (x2 (t), t) ✤➲✉ ♥➡♠ tr♦♥❣ Rm+1 ✈ỵ✐ ♠å✐ t ∈ I ✱ t❤➻ x1 (t) = x2 (t), ∀t ∈ I ✳ ✺✷ ... g = 4xi −yi , ui+1 = xi +0.1f, vi+ 1 = yi + 0.1g}; ✷✽ {f1 = Simplif y[F unction[{x, y}, 3x − 2y][ui+1 , vi+ 1 ]], g1 = Simplif y[F unction[{x, y}, 4x − y][ui+1 , vi+ 1 ]]}; {P rint[”(”, Simplif... − 3yi − 2, ui+1 = xi + 0.1f, vi+ 1 = yi + 0.1g}; {f1 = Simplif y[F unction[{x, y}, x − y][ui+1 , vi+ 1 ]], ✸✷ g1 = Simplif y[F unction[{x, y}, 5x − 3y − 2][ui+1 , vi+ 1 ]]}; {P rint[”(”, Simplif... t✉②➳♥ t➼♥❤ ✈➔ → t✉②➳♥ t➼♥❤ ❝â ✈à tr➼ q✉❛♥ trå♥❣ ✈➔ ✤â♥❣ ✈❛✐ trá t❤❡♥ ❝❤èt tr♦♥❣ ✈✐➺❝ ♠æ t↔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ tổ q ỗ t õ ✤÷đ❝ ✈❛✐ trá q✉❛♥ trå♥❣ ✤â ✈➔ ✤÷đ❝ sü ✤ë♥❣ ủ ỵ t ữợ ❚❙✳ ▲➯ ❍↔✐ ❚r✉♥❣