Ứng dụng phần mềm mathematica cho phương pháp runge kutta

✣❸■ ❍➴❈ ✣⑨ ◆➂◆● ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❑❍❖❆ ❚❖⑩◆ ✖✖✖✖✖ ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P Ù◆● ❉Ö◆● P❍❺◆ ▼➋▼ Pì PP ữợ ▲➊ ❍❷■ ❚❘❯◆● ❙✐♥❤ ✈✐➯♥ t❤ü❝ ❤✐➺♥✿ P❍❸▼ ❚❍➚ ❍■➋◆ ✣➔ ◆➤♥❣✱ ✵✺✴✷✵✶✺ ▼ö❝ ❧ö❝ ▲❮■ ❈❷▼ ❒◆ ▼Ð ✣❺❯ ✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ✸ ✹ ✻ ✶✳✶ ❈→❝ ❦❤→✐ ♥✐➺♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ tỗ t t t ✳ ✼ ✷ P❍×❒◆● P❍⑩P ❘❯◆●❊ ✲ ❑❯❚❚❆ ✣➮■ ❱❰■ ❈➷◆● ❚❍Ù❈ ❳❻P ❳➓ ❇❾❈ ❇➮◆ ❘❑ ✶✵ ✷✳✶ P❤÷ì♥❣ ♣❤→♣ t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✷ P❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡✲❑✉tt❛ ✲ ❈ỉ♥❣ t❤ù❝ ①➜♣ ①➾ ❜➟❝ ❜è♥ Pữỡ tt ố ợ P❚❱P ❝➜♣ ♥ ✭♥❃✶✮ ✳ ✶✺ ✷✳✹ ×✉ ✤✐➸♠✱ ❤↕♥ ❝❤➳ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡ ✲ ❑✉tt❛ ✳ ✳ ✳ ✳ ✶✽ ✷✳✺ Ù♥❣ ❞ư♥❣ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡ ✲ ❑✉tt❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✸ Ù◆● ❉Ư◆● ▼❆❚❍❊▼❆❚■❈❆ ❈❍❖ P❍×❒◆● P❍⑩P ❘❯◆●❊ ✲ ❑❯❚❚❆ ✷✵ ✸✳✶ ❙ü r❛ ✤í✐ ✈➔ ♣❤→t tr✐➸♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ữỡ tr ợ tt ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ❱⑨■ ◆➆❚ ❱➋ ❈❆❘▲ ❘❯◆●❊ ❱⑨ ❲■▲❍❊▲▼ ❑❯❚❚❆ ❑➌❚ ▲❯❾◆ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✷ ✸✷ ✸✸ ✸✹ ▲❮■ ❈❷▼ ❒◆ ❚❙✳ ▲➯ ❍↔✐ ❚r✉♥❣ ❙❛✉ ♠ët t❤í✐ ❣✐❛♥ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ự ữợ sỹ ữợ sỹ t t➻♥❤ ❝õ❛ t❤➛② ❣✐→♦ ✱ ✤➳♥ ♥❛② ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❝õ❛ ❡♠ ✤➣ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤✳ ❊♠ ①✐♥ ❜➔② tä sü ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ✤➳♥ ❇❛♥ ●✐→♠ ❍✐➺✉ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ P❤↕♠ ✲ ✣↕✐ ❍å❝ ✣➔ ◆➤♥❣✱ ❇❛♥ ❝❤õ ♥❤✐➺♠ ❦❤♦❛ ❚♦→♥✱ ✤➣ t↕♦ ❝ì ❤ë✐ ❝❤♦ ❡♠ ✤÷đ❝ ❧➔♠ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣✳ ❈❤ó♥❣ ❡♠ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ s➙✉ s➢❝ ✤➳♥ t➜t ❝↔ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ tr♦♥❣ tr÷í♥❣✱ ✤➦❝ ❜✐➺t ❧➔ ❝→❝ t❤➛② ❝æ ❣✐→♦ tr♦♥❣ ❦❤♦❛ ❚♦→♥ ✤➣ t➟♥ t➻♥❤ ❝❤➾ ❞↕②✱ tr✉②➲♥ ✤↕t ❝❤♦ ❡♠ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❜ê ➼❝❤ ✈➔ qỵ tr sốt tớ ứ q ì♥ sü ❣✐ó♣ ✤ï✱ ❝❤✐❛ s➫ ❝õ❛ t➜t ❝↔ ❝→❝ ❜↕♥ tr♦♥❣ ❧ỵ♣ tr♦♥❣ t❤í✐ ❣✐❛♥ ❡♠ ❧➔♠ ❦❤â❛ ❧✉➟♥✳ ❈✉è✐ ❝ị♥❣✱ ❡♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ t❤➛② ▲➯ r ữớ trỹ t ữợ ❧✉æ♥ q✉❛♥ t➙♠✱ ✤ë♥❣ ✈✐➯♥ ❝❤➾ ❞➝♥ t➟♥ t➻♥❤ ✤➸ ❝❤ó♥❣ ❡♠ ❤♦➔♥ t❤➔♥❤ tèt ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ♥➔②✳ ❚✉② ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣ s♦♥❣ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✈➝♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât ✈➲ ♥ë✐ ❞✉♥❣ ❧➝♥ ❤➻♥❤ t❤ù❝ tr➻♥❤ ❜➔②✱ ❝❤ó♥❣ ❡♠ r➜t ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ sü ✤â♥❣ õ qỵ t ổ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✺ ❙✐♥❤ ✈✐➯♥ t❤ü❝ ❤✐➺♥ P❤↕♠ ❚❤à é ỵ ỹ t➔✐ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❧➔ ♠ỉ ❤➻♥❤ ♠ỉ t↔ ❦❤→ tèt q✉→ tr➻♥❤ ❝❤✉②➸♥ ✤ë♥❣ tr♦♥❣ tü ♥❤✐➯♥ ✈➔ ❦➽ t❤✉➟t✳ ❚r♦♥❣ ❧➽♥❤ ✈ü❝ t♦→♥ ù♥❣ ❞ư♥❣ t❤÷í♥❣ ❣➦♣ rt t q tợ ữỡ tr ♣❤➙♥ t❤÷í♥❣✳ ❱➻ ✈➟②✱ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ✤â♥❣ ♠ët ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣ ❧➼ t❤✉②➳t t♦→♥ ❤å❝✳ ❚r♦♥❣ ✤↕✐ ✤❛ sè tr÷í♥❣ ❤đ♣ ♥❤÷✿ ❝→❝ ❜➔✐ t♦→♥ ❝â ❤➺ sè ❜✐➳♥ t❤✐➯♥✱ ❝→❝ ❜➔✐ t♦→♥ ♣❤✐ t✉②➳♥✱ ❝→❝ ❜➔✐ t♦→♥ tr➯♥ ♠✐➲♥ ❜➜t ❦➻✱ ✳✳✳ t❤➻ ♥❣❤✐➺♠ t÷í♥❣ ♠✐♥❤ ❤♦➦❝ ❦❤ỉ♥❣ ❝â ❤♦➦❝ ♥➳✉ ❝â t❤➻ r➜t ♣❤ù❝ t↕♣ ✈➔ ✈✐➺❝ ❦❤↔♦ s→t t➼♥❤ ❝❤➜t ♥❣❤✐➺♠ ❣➦♣ r➜t ♥❤✐➲✉ ❦❤â ❦❤➠♥✱ ❞♦ ✤â ♣❤↔✐ t➻♠ ✤➳♥ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ✤➸ t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣✳ ❉♦ ♥❤✉ ❝➛✉ t❤ü❝ t✐➵♥ ❝ị♥❣ ✈ỵ✐ sü ♣❤→t tr✐➸♥ ❝õ❛ ♠➻♥❤✱ tr♦♥❣ t♦→♥ ❤å❝ ①✉➜t ❤✐➺♥ ♥❤✐➲✉ ♣❤÷ì♥❣ ♣❤→♣ ✤➸ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣✿ P❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ t➼❝❤✱ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✉é✐ ❚❛②❧♦r✱ ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ❧✐➯♥ t✐➳♣ P✐❝❛r❞✱ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ sè ♥❤÷✿ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r✱ ❊✉❧❡r ❝↔✐ t✐➳♥✱ ❆❞❛♠✱ ♣❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡✲❑✉tt❛✱ ✳✳✳ ✳ ▼❛t❤❡♠❛t✐❝❛ ❧➔ ❝ỉ♥❣ ❝ư ❧➟♣ tr➻♥❤ ♠↕♥❤ ✈ỵ✐ ❤ì♥ ✼✵✵ ❤➔♠ ❝â s➤♥ tr♦♥❣ t❤ü ✈✐➺♥ ❤➔♠✳❱✐➺❝ sû ❞ư♥❣ ▼❛t❤❡♠❛t✐❝❛ ♥❤❛♥❤ ❣➜♣ ♥❤✐➲✉ ❧➛♥ s♦ ✈ỵ✐ ❣✐↔✐ ❜➡♥❣ t❛② t❤ỉ♥❣ t❤÷í♥❣ ❜➡♥❣ ❝→❝❤ ù♥❣ ❞ư♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣✳ ❚r♦♥❣ ♣❤↕♠ ✈✐ ✈➔ ②➯✉ ❝➛✉ ❝õ❛ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ ❝❤ó♥❣ ❡♠ s➩ tr➻♥❤ ❜➔② ù♥❣ ❞ư♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ❝❤♦ ♣❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡✲ ✷✳ ▼ư❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ ❑✉tt❛ ✤➸ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ♣❤÷ì♥❣ tr➻♥❤ ợ t ữỡ tt ởt tr♦♥❣ ♥❤ú♥❣ ♣❤÷ì♥❣ ♣❤→♣ r➜t ❤❛② ✤➸ t➻♠ ♥❣❤✐➺♠ ❣➛♥ ú ữỡ tr tữớ ợ ①→❝ ❝❛♦✱ ✈➔ t✐➳t ❦✐➺♠ t❤í✐ ❣✐❛♥✳ ✹ ●✐ỵ✐ t❤✐➺✉ ♠ët ♥❣ỉ♥ ♥❣ú ❧➟♣ tr➻♥❤ ♠↕♥❤✱ ▼❛t❤❡♠❛t✐❝❛✱ ❤é trđ ❝❤♦ ❣✐↔♥❣ ✈✐➯♥ ✈➔ s✐♥❤ ✈✐➯♥ tr♦♥❣ ✈✐➺❝ ❣✐↔♥❣ ❞↕② ✈➔ ❤å❝ t➟♣ ●✐↔✐ t➼❝❤ sè ♥â✐ ❝❤✉♥❣ ✈➔ ✸✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ✈✐➺❝ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ♥â✐ r✐➯♥❣✳ ❚❤❛♠ ❦❤↔♦ ✈➔ ❞à❝❤ ❝→❝ t➔✐ ❧✐➺✉ t✐➳♥❣ ❆♥❤✳ ❚ê♥❣ ❤đ♣ ✈➔ tr➻♥❤ ❜➔②✳ ✺ ❈❤÷ì♥❣ ✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ✶✳✶ ❈→❝ ❦❤→✐ ♥✐➺♠ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❝â ❞↕♥❣✿ F (x, y, y, ˙ yă, , y (n) ) = 0, tr♦♥❣ ✤â ✭✶✳✶✮ ②❂②✭①✮ ❧➔ ➞♥ ❤➔♠ ❝➛♥ t➻♠ ✈➔ ♥❤➜t t❤✐➳t ♣❤↔✐ ❝â sü t❤❛♠ ❣✐❛ ❝õ❛ ✤↕♦ ❤➔♠ ✭✤➳♥ ❝➜♣ ♥➔♦ ✤â✮ ❝õ❛ ➞♥✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ➞♥ ❤➔♠ ❝➛♥ t➻♠ ❧➔ ❤➔♠ ♥❤✐➲✉ ❜✐➳♥ ✭①✉➜t ❤✐➺♥ ❝→❝ ✤↕♦ ❤➔♠ r✐➯♥❣✮ t❤➻ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝á♥ ❣å✐ ❧➔ r✐➯♥❣✳ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ ❚❤ỉ♥❣ t❤÷í♥❣ t❛ ①➨t ữỡ tr ợ số ởt ❜✐➳♥ ②❂②✭①✮ ①→❝ ✤à♥❤ tr➯♥ ❦❤♦↔♥❣ ♠ð I ⊂ R ✱ ❦❤✐ ✤â ❤➔♠ ❋ tr♦♥❣ ✤➥♥❣ n+1 t❤ù❝ tr➯♥ ①→❝ ✤à♥❤ tr♦♥❣ ♠ët t➟♣ ♠ð ● ❝õ❛ R × R ✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ t❤ü❝ ➞♥ ❤➔♠ ❝➛♥ t➻♠ ❧➔ ✈❡❝t♦r ✲ ❤➔♠ ✭❤➔♠ ✈ỵ✐ ❣✐→ trà ✈❡❝t♦r✮ ❋ ❧➔ ♠ët →♥❤ ①↕ ♥❤➟♥ ❣✐→ trà tr♦♥❣ Rm ✱ ❦❤✐ ✤â ✭✶✳✶✮ ✤÷đ❝ ❤✐➸✉ ❧➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✳ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ❝➜♣ ■ ❝â ❞↕♥❣ tê♥❣ q✉→t✿ y(x) = (y1 (x), , ym (x))T ∈ Rm ✱ F (x, y, y) ˙ = 0, tr♦♥❣ ✤â ❋ ①→❝ ✤à♥❤ tr♦♥❣ ♠✐➲♥ G ∈ R✱ ✭✶✳✷✮ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ■ õ t t ữợ ữủ t❤❡♦ ✤↕♦ ❤➔♠✮ y˙ = f (x, y), ✈ỵ✐ ❢ ❧✐➯♥ tö❝ tr♦♥❣ ♠✐➲♥ D ⊂ R2 ✳ ✻ ✭✶✳✸✮ ỵ tỗ t t t t ố ợ ữỡ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ♠ët ❚➻♠ ♥❣❤✐➺♠ ②✭①✮ t❤ä❛✿ y˙ = f (x, y), y(x0 ) = y0 tr♦♥❣ ✤â (x0 , y0 ) ∈ D ✈➔ y(x0 ) = y0 ✭✶✳✹✮ ✤÷đ❝ ❣å✐ ❧➔ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉✳ ❚❛ t t ố ợ ữỡ tr ữủ ố ợ tr♦♥❣ ✤â ●✐↔ sû ②✭①✮ ❢ ①→❝ ✤à♥❤ ✈➔ ❧✐➯♥ tö❝ tr➯♥ ♠✐➲♥ ♠ð D ⊂ R2 ✳ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭✶✳✹✮✱ t➼❝❤ ♣❤➙♥ ✷ ✈➳ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ tr♦♥❣ ✭✶✳✹✮ t❛ ✤÷đ❝ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ✤è✐ ✈ỵ✐ ②✭①✮ ❧➔✿ x f (t, y(t))dt, y(x) = y0 + ✭✶✳✺✮ x0 ❘ã r➔♥❣ ♠é✐ ♥❣❤✐➺♠ ❝õ❛ ✭✶✳✹✮ ❝ô♥❣ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✶✳✺✮ ✈➔ ♥❣÷đ❝ ❧↕✐✱ ♠é✐ ♥❣❤✐➺♠ ❝õ❛ ✭✶✳✺✮ ✤➲✉ ❦❤↔ ✈✐ ✈➔ ❧✐➯♥ tö❝ tr➯♥ ♠ët ❦❤♦↔♥❣ ■ ♥➔♦ ✤â ✈➔ t❤ä❛ ✭✶✳✹✮✳ ❇ê ✤➲ ✶✳✶✳ ●✐↔ sû ❢✭①✱②✮ ❧✐➯♥ tö❝ tr➯♥ ❤➻♥❤ ❝❤ú ♥❤➟t D = {(x, y) ∈ R2 , |x − x0 | ≤ a, |y − y0 | ≤ b}, ✣➦t M = ♠❛① ⑤❢✭①✱②✮⑤✱ ❤❂ ♠✐♥ {a, b/M }✱ ✭t ❛ ❝â t❤➸ ❣✐↔ sû M = 0✮✳ ❚❛ ✤➦t✿ y0(x) = y0, ∀x ∈ [x0 − h, x0 + h], x y1 (x) = y0 + f (t, y0 (t))dt, x0 x f (t, yn−1 (t))dt, n ≥ yn (x) = y0 + x0 ❉➣② yn(x) ✤÷đ❝ ❣å✐ ❧➔ ❞➣② Pr ỹ tỗ t t ♥❣❤✐➺♠ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ✭✣✐➲✉ ❦✐➺♥ ▲✐♣s❝❤✐t③✮ ❢✭①✱②✮ ①→❝ ✤à♥❤ tr➯♥ ♠✐➲♥ D ⊂ R✳ ❚❛ ♥â✐ ❢ t❤ä❛ ✤✐➲✉ st t tr tỗ t ❤➡♥❣ sè ❞÷ì♥❣ ▲ ✭❣å✐ ❧➔ ❤➡♥❣ sè ▲✐♣s❝❤✐t③✮ s❛♦ ❝❤♦✿ ❈❤♦ ❤➔♠ |f (x, y1 ) − f (x, y2 )| ≤ L|y1 − y2 |✱ ✈ỵ✐ ♠å✐ ✼ (x, y1 ), (x, y2 ) ∈ D ✣à♥❤ ỵ sỷ tr tử ✈➔ t❤ä❛ ✤✐➲✉ ❦✐➺♥ ▲✐♣s❝❤✐t③ t❤❡♦ ❜✐➳♥ ② tr➯♥ ❤➻♥❤ ỳ t ỵ tỗ t t D = {(x, y) ∈ R2 , |x − x0 | ≤ a, |y − y0 | ≤ b}, ❑❤✐ õ t tỗ t ✈➔ ❞✉② ♥❤➜t tr➯♥ ✤♦↕♥ I := [x0 − h, x0 + h], ✈ỵ✐ ❤❂ ♠✐♥ {a, b/M } ✈➔ ự ỹ tỗ t ❚❛ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ♣❤➨♣ ❧➦♣ P✐❝❛r❞ ❤ë✐ tö ✤➲✉ tr➯♥ ■ ✤➳♥ ♠ët ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❈❛✉❝❤②✳ ✣➛✉ t✐➯♥ ❜➡♥❣ q✉② ♥↕♣ t❛ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣✿ |yk+1 (x) − yk (x)| ≤ M Lk ✈ỵ✐ ♠å✐ |x − x0 |k+1 , (k + 1)! x ∈ I✳ ❦ ❂✵ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ trð t❤➔♥❤ | xx f (t, y0(t))dt| ≤ M |x − x0|, ✈ỵ✐ x0 ≤ x✱ ✈➻ ▼❂ ♠❛① ⑤❢✭①✱②✮ ⑤ s✉② r❛ f (x, y) ≤ M ✱ t❛ ❝â ❱ỵ✐ x x f (t, y0 (t))dt ≤ |f (t, y0 (t))|dt, x0 x0 x ⇒| f (t, y0 (t))dt| ≤ M (x − x0 ) ≤ M |x − x0 |, x0 ✈➟② ❜➜t ✤➥♥❣ t❤ù❝ ✤ó♥❣✳ k − 1✱ ❦❤✐ ✤â ✈ỵ✐ x0 ≤ x ≤ x0 + h x |yk+1 (x) − yk (x)| = | x0 [f (t, yk (t)) − f (t, yk−1 (t))]dt| ≤ ●✐↔ sû t❛ ❝â ✤✐➲✉ ❦✐➺♥ ✤â ✈ỵ✐ ≤| x x0 [f (t, yk (t)) − f (t, yk−1 (t))]dt| ≤ L ≤ M Lk x x0 x x0 |yk (t) − yk−1 (t)|dt ≤ (x − x0 )k /k!dt = M Lk (x − x0 )(k+1) /(k + 1)! x0 − h ≤ x ≤ x0 t❛ ❝❤ù♥❣ ♠✐♥❤ ❳➨t ❞➣② ❤➔♠ yk (x) tr t õ ợ tữỡ tỹ |yk+p (x) − yk (x)| ≤ |yk+p (x) − yk+p−1 (x)| + |yk+p−1 (x) − yk+p−2 (x)|+ + |yk+1 (x) − yk (x)| ≤ ≤ M L j≥k+1 M (L|x−x0 |)k+p L{ (k+p)! (Lh)j j! ✳ ✽ + + (L|x−x0 |)k+1 } (k+1)! ≤ t❛ ❝â ❈❤✉é✐ sè ∞ (Lh)j ❧➔ ❤ë✐ tư✱ ♥➯♥ ♣❤➛♥ ❞÷ ❝õ❛ ♥â ❝â t❤➸ ❧➔♠ ❝❤♦ ❜➨ j=0 j! ❦ ✤õ ❧ỵ♥✳ ❚❤❡♦ t✐➯✉ ❝❤✉➞♥ ❈❛✉❝❤②✱ ❞➣② {yk (x)} ❤ë✐ tư ✤➲✉ tr➯♥ ■ ✤➳♥ ❤➔♠ ②✭①✮✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ②✭①✮ ❧➔ ♥❣❤✐➺♠ t❛ ❝❤➾ ❝➛♥ q✉❛ ❣✐ỵ✐ ❤↕♥ tr♦♥❣ tị② þ ❦❤✐ ✤➥♥❣ t❤ù❝ x yk+1 (x) = y0 + ❱➻ ❞➣② ❤➔♠ yk (x) ❤ë✐ tö ✤➲✉✱ ❢ ❧✐➯♥ tö❝ tr➯♥ ❤➻♥❤ ❝❤ú ♥❤➟t ❉ ♥➯♥ ❞➣② ❤➔♠ ❢✭t✱②✭t✮✮ ❞♦ ✤â ❝â t❤➸ ❝❤✉②➸♥ ❣✐ỵ✐ ❤↕♥ q✉❛ ❞➜✉ t➼❝❤ ♣❤➙♥ ✤➸ ✤÷đ❝ ✤➥♥❣ t❤ù❝ ✭✶✳✺✮✳ ❱➟② ②✭①✮ ❧➔ ♥❣❤✐➺♠ ❝õ❛ {f (t, yk (t))} ■ f (t, yk (t))dt, x0 ❤ë✐ tö ✤➲✉ tr➯♥ ✤➳♥ ❤➔♠ ❜➔✐ t♦→♥ ❈❛✉❝❤② ✭✶✳✹✮✳ ❜✮✳ ❚➼♥❤ ❞✉② ♥❤➜t✳ ●✐↔ sû ❜➔✐ t♦→♥ ❈❛✉❝❤② ❝á♥ ❝â ♥❣❤✐➺♠ t❛ ❝â ③✭①✮ ❦❤✐ ✤â x y(x) − z(x) = [f (t, y(t)) − f (t, z(t))]dt x0 ❙✉② r❛ x |y(x) − z(x)| = | [f (t, y(t)) − f (t, z(t))]dt| ≤ 2M |x − x0 | x0 ❚ø ✤â x |y(x) − z(x)| = | x [f (t, y(t)) − f (t, z(t))]dt| ≤ L x0 |y(t) − z(t)|dt x0 0) ≤ 2M L (x−x ▲➦♣ ❧↕✐ q✉→ tr➻♥❤ tr➯♥✱ t❛ ự ữủ r ợ số tỹ k→∞ t❤➻ |y(x) − z(x)| ≤ 2M Lk |x − x0 |k+1 , x ∈ I (k + 1)! |y(x) − z(x)| = ■ ②✭①✮ ❧➔ ❞✉② ♥❤➜t✳ tr➯♥ ✾ ⇒ y(x) ≡ z(x) ❦ ◆❤÷ ✈➟② ♥❣❤✐➺♠ ❈❤÷ì♥❣ ✷ P❍×❒◆● P❍⑩P ❘❯◆●❊ ✲ ❑❯❚❚❆ ✣➮■ ❱❰■ ❈➷◆● ❚❍Ù❈ ❳❻P ❳➓ ❇❾❈ ❇➮◆ ❘❑ ✷✳✶ P❤÷ì♥❣ ♣❤→♣ t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ✣➸ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ t❛ ❝â t❤➸ ❣✐↔✐ t❤❡♦ ❤❛✐ ♣❤÷ì♥❣ ♣❤→♣✿ P❤÷ì♥❣ ♣❤→♣ t➻♠ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝✿ ❇➡♥❣ ❝→❝❤ ❞ü❛ ✈➔♦ ❝→❝❤ t➼♥❤ t➼❝❤ ♣❤➙♥ trü❝ t✐➳♣✱ ①→❝ ✤à♥❤ ❝→❝ ❞↕♥❣ tê♥❣ q✉→t ❝õ❛ rỗ ỹ ✤à♥❤ ♥❣❤✐➺♠ r✐➯♥❣ ❝➛♥ t➻♠✳ P❤÷ì♥❣ ♣❤→♣ t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣✿ ❳✉➜t ♣❤→t tø ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉✱ ♣❤÷ì♥❣ ♣❤→♣ õ t ởt ợ ữỡ tr rở ỡ rt s ợ ữỡ ♣❤→♣ trü❝ t✐➳♣✱ ❞♦ ✤â ♣❤÷ì♥❣ ♣❤→♣ ♥➔② ✤÷đ❝ ❞ị♥❣ ♥❤✐➲✉ tr♦♥❣ t❤ü❝ t➳✳ ❚r♦♥❣ t❤ü❝ t➳ ❝â ♥❤✐➲✉ ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ♥❤÷✿ P❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r✱ ❊✉❧❡r ❝↔✐ t✐➳♥✱ ❘✉♥❣❡ ✲ ❑✉tt❛✳✳✳ ✳ ❚r♦♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✤â t❤➻ ♣❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡ ✲ ❑✉tt❛ ✤÷đ❝ ❞ị♥❣ ♥❤✐➲✉ tr♦♥❣ t❤ü❝ t➳ ✈➻ t➼♥❤ ❤✐➺✉ q✉↔ ♠➔ ♥â ♠❛♥❣ ợ Pữỡ tt ❞♦ ❤❛✐ ♥❤➔ ❜→❝ ❤å❝ ♥❣÷í✐ ✣ù❝ ❧➔ ❈❛r❧ ❘✉♥❣❡ ✈➔ ❲✐❧❤❡❧♠ ❑✉tt❛ ✤÷❛ r❛✳ ❚❛ ❧✉ỉ♥ ❣✐↔ t❤✐➳t ❜➔✐ t♦→♥ ✤➦t r❛ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t ✈➔ ♥❣❤✐➺♠ ✤â ✤õ trì♥✱ ♥❣❤➽❛ ❧➔ ♥â ❝â ✤↕♦ ❤➔♠ ✤➳♥ ❝➜♣ ữỡ ệ Pì P❍⑩P ❘❯◆●❊ ✲ ❑❯❚❚❆ ✸✳✶ ❙ü r❛ ✤í✐ ✈➔ ♣❤→t tr✐➸♥ ❚r♦♥❣ ❝→❝ ♠ỉ♥ ❤å❝ ù♥❣ ❞ư♥❣ ❝➛♥ ❣✐↔✐ q✉②➳t ❝→❝ ❜➔✐ t➼♥❤ t♦→♥ ❝ư t❤➸ ✈ỵ✐ t❤í✐ ❣✐❛♥ ♥❤❛♥❤ ♥❤➜t ❧➔ ②➯✉ ❝➛✉ ❝➜♣ t❤✐➳t✳ ▼❛t❤❡♠❛t✐❝❛ ❧➔ ♠ët ❝æ♥❣ ❝ư ❧➟♣ tr➻♥❤ ♠↕♥❤ ✈ỵ✐ ❤ì♥ ✼✵✵ ❤➔♠ ❝â s➤♥ tr♦♥❣ t❤÷ ✈✐➺♥ ❤➔♠✱ t❤ü❝ ❤✐➺♥ ♥❤✐➲✉ ❝❤ù❝ ♥➠♥❣ ❦❤→❝ ♥❤❛✉✳ P❤✐➯♥ ❜↔♥ ✤➛✉ t✐➯♥ ❝õ❛ ▼❛t❤❡♠❛t✐❝❛ ✤÷đ❝ ❤➣♥❣ ❲♦❧❢r❛♠ ❘❡s❡❛r❝❤ ♣❤→t ❤➔♥❤ ✈➔♦ ♥➠♠ ✶✾✽✽✳ ✣➙② ❧➔ ❤➺ t❤è♥❣ ♣❤➛♥ ♠➲♠ ♥❤➡♠ t❤ü❝ ❤✐➺♥ ❝→❝ t➼♥❤ t♦→♥ tr➯♥ ▼→② t➼♥❤ ✤✐➺♥ tû✳ ◆â ❧➔ tê ❤ñ♣ ❝→❝ t➼♥❤ t♦→♥ t t số ỗ t ✈➔ ♥❣æ♥ ♥❣ú ❧➟♣ tr➻♥❤ ✈✐ t➼♥❤✳ ❈æ♥❣ tr➻♥❤ ♥➔② ✤÷đ❝ ①❡♠ ❧➔ t❤➔♥❤ tü✉ ❝❤➼♥❤ tr♦♥❣ ❧➽♥❤ ✈ü❝ ❦❤♦❛ ❤å❝ t➼♥❤ t♦→♥✳ ▼❛t❤❡♠❛t✐❝❛ ❧➔ ♥❣ỉ♥ ♥❣ú t➼❝❤ ❤đ♣ ✤➛② ✤õ ♥❤➜t ❝→❝ t➼♥❤ t♦→♥ ❦ÿ t❤✉➟t✳ ▲➔ ❞↕♥❣ ♥❣æ♥ ỳ ỹ tr ỵ ỷ ỵ ỳ ❧✐➺✉ t÷ì♥❣ ù♥❣✳ ❚❤➳ ❤➺ ♥❣ỉ♥ ♥❣ú ❣✐↔✐ t➼❝❤ ✤➛✉ t✐➯♥ ✤â ❧➔✿ ▼❛❝s②♠❛✱ ❘❡❞✉❝❡✱ ✳✳✳ r❛ ✤í✐ tø ♥❤ú♥❣ ♥➠♠ ✻✵ ❝õ❛ t❤➳ ❦➾ ❳❳✳ ❈→❝ ♥❣æ♥ ♥❣ú ♥➔② ũ t t ỵ ữủ ❝❛♦✳ ◆❤÷đ❝ ✤✐➸♠ ❝õ❛ ❝❤ó♥❣ ❧➔ ❝❤õ ②➳✉ ✤÷đ❝ ✤à♥❤ ữợ tr t ợ t t ❧➔ ❝→❝ ♥❣æ♥ ♥❣ú✿ ▼❛♣❧❡✱ ▼❛t❤❧❛❜✱ ▼❛t❤❡♠❛t✐❝❛✳ ❈→❝ ♥❣æ♥ ♥❣ú ♥➔② ❝â ÷✉ ✤✐➸♠ ❧➔ ❝❤↕② ♥❤❛♥❤ ❤ì♥✱ ✈➔ ❝❤➜♣ ♥❤➟♥ ❜ë ♥❤ỵ ♥❤ä ✷✵ ❤ì♥✱ ❝❤↕② ❤♦➔♥ ❤↔♦ tr➯♥ ❝→❝ ♠→② t➼♥❤ ❝→ ♥❤➙♥✳ ❚r♦♥❣ ❝→❝ ♥❣æ♥ ♥❣ú t➼♥❤ t♦→♥ ❧♦↕✐ ♥➔② ♥ê✐ ❜➟t ❧➔ ♥❣ỉ♥ ♥❣ú ▼❛t❤❡♠❛t✐❝❛✱ ✈ỵ✐ ÷✉ ✤✐➸♠ ✈÷ñt trë✐ ✈➲ ❣✐❛♦ ❞✐➺♥ t❤➙♥ t❤✐➺♥✱ ✈➲ ỗ t s t ỷ ỵ ỳ ❧✐➺✉ ❦❤æ♥❣ t❤✉❛ ❦➨♠ ❝→❝ ♥❣æ♥ ♥❣ú t➼♥❤ t♦→♥ ❦❤→❝✳ ◆❤í ✈➔♦ ❦❤↔ ♥➠♥❣ ♠ỉ ❤➻♥❤ ❤â❛ ✈➔ ♠ỉ ♣❤ä♥❣ ❝→❝ ❤➺ ❧ỵ♥ ❦➸ ❝↔ ❝→❝ ❤➺ ✤ë♥❣ ♠➔ ▼❛t❤❡♠❛t✐❝❛ ổ ữủ ự tr ỹ t ỵ ❦ÿ t❤✉➟t ✈➔ t♦→♥ ♠➔ ❝á♥ ✤÷đ❝ ♠ð rë♥❣ ù♥❣ ❞ö♥❣ tr♦♥❣ ✈→❝ ❧➽♥❤ ✈ü❝ ❙✐♥❤ ❤å❝ ✈➔ ❝→❝ ❦❤♦❛ ❤å❝ ❦❤→❝✳ ▼❛t❤❡♠❛t✐❝❛ ❝â ♥❤✐➲✉ ❱❡rs✐♦♥ ❞♦ ❧✉ỉ♥ ❧✐➯♥ tư❝ ✤÷đ❝ ❝↔✐ t✐➳♥ ✈➔ ❤♦➔♥ t❤✐➺♥✿ ✶✳✷✱ ✷✳✵✱ ✷✳✷✱ ✸✳✵✱ ✹✳✵✱ ✹✳✷✱ ✳✳✳ P❤✐➯♥ ❜↔♥ ♠ỵ✐ ♥❤➜t ❤✐➺♥ ♥❛② ❝õ❛ ▼❛t❤❡♠❛t✐❝❛ ❧➔ ✼✳✵ ✳ ▼❛t❤❡♠❛t✐❝❛ ❝✉♥❣ ❝➜♣ r➜t ♥❤✐➲✉ ❝❤ù❝ ♥➠♥❣✱ ♠ët sè ❝❤ù❝ ♥➠♥❣ t❤ỉ♥❣ ❞ư♥❣ ❧➔✿ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✱ ❣✐↔✐ ♠❛ tr➟♥✱ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤✱ ✈➩ ỗ t tt ữủ sû ❞ư♥❣ ✈➔ ❣✐↔♥❣ ❞↕② t↕✐ ♥❤✐➲✉ tr÷í♥❣ ❈❛♦ ✤➥♥❣✱ ✣↕✐ ❤å❝✱ ✳✳✳ ✤➙② ❧➔ ❝ỉ♥❣ ❝ư ❤é trđ tr♦♥❣ ợ ữỡ ổ ữỡ tr ợ tt P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ♠ët ❱➼ ❞ư ✸✳✶ ✐✮ y˙ = y − x − 1; y(0) = ✰✮ ❚➻♠ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ✈➔ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ❜➡♥❣ ữỡ ố ợ x [0, 1]; h = 0, ❚❛ ❝â✿ y˙ = y − x − 1, ⇔ y˙ − y = −x − ✣➙② ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ✶ ✈ỵ✐✿ q(x) = −x − 1; ◆❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❝â ❞↕♥❣✿ ✷✶ p(x) = −1✱ y = e− p(x)dx [ q(x)e p(x)dx = −dx = −x; q(x)e p(x)dx dx = (−x − 1)e−x dx = = xe−x + 2e−x p(x)dx dx + C] ❚❛ ❝â✿ −xe−x dx − e−x dx = ⇒ y = ex [xe−x + 2e−x + C] = x + + Cex y0 = ⇒ C = −1 ◆❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❧➔✿ y = x + − ex ⑩♣ ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ ❘❑✹ ✤➸ t➻♠ ❝→❝ ❣✐→ trà x = 0, 5; ợ ợ y1 , y2 , tữỡ ự ✈ỵ✐ ❤ ❂✵✱✺✱ ✤➦t✿ f (x, y) = y − x − 1✳ x1 = 0, : k1 = hf (x0 , y0 ) = 0, 5(1 − − 1) = 0, 1 k2 = 0, 5[1 + − (0 + 0, 5) − 1] = −0, 125, 2 1 k3 = 0, 5[1 + (−0, 125) − (0 + 0, 5) − 1] = −0, 15625, 2 k4 = 0, 5[1 − 0, 15625 − (0 + 0, 5) − 1] = −0, 32813 ⇒ y1 = + [0 + 2(−0, 125) + 2(−0, 15625) − 0, 32813] = 0, 85156 ✯✮ ❱ỵ✐ x2 = : k1 = 0, 5(0, 85156 − 0, − 1), 1 k2 = 0, 5[0, 85156 + (−0, 32422) − (0, + 0, 5) − 1] = −0, 532028, 2 1 k3 = 0, 5[0, 85156 + (−0, 532028) − (0, + 0, 5) − 1] = −0, 58179, 2 k4 = 0, 5[0, 85156 − 0, 58179 − (0, + 0, 5) − 1] = −0, 86512 ⇒ y2 = 0, 85156 + [−0, 32422 + 2(−0, 53028) + 2(−0, 58179) − 0, 86512] = 0, 28231 ✰✮ ❙û ❞ö♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ✭❣✐↔✐ trü❝ t✐➳♣✮ ■♥[1] := DSolve[{y[x] ˙ == y[x] − x − 1, y[0] == 1}, y, x] ✷✷ Out[1] = {{y → F unction[{x}, − ex + x]}} ■♥[2] := u := − ex + x ■♥[3] := x = 0.5 ■♥[6] := x = Out[3] = x = 0.5 Out[6] = ■♥[4] := u ■♥[7] := u Out[4] = 2.5 − e0.5 Out[7] = − e ■♥[5] := N [2.5 − e0.5] ■♥[8] := N [3 − e] Out[5] = 0.851279 Out[8] = 0.28172 ❚ø ✤â t❛ ❝â ❜↔♥❣ ❣✐→ trà s❛✉✿ ① ❈ỉ♥❣ t❤ù❝ ❘❑ ✈ỵ✐ ❤ ❂✵✱✺ ●✐→ trà ❝❤➼♥❤ ①→❝ ❣✐↔✐ ❜➡♥❣ ♠→② t➼♥❤ ✵ ✶ ✶ ✵✱✺ ✵✱✽✺✶✺✻ ✵✱✽✺✶✷✽ ✶ ✵✱✷✽✷✸✶ ✵✱✷✽✶✼✷ ❇↔♥❣ ✸✳✶✿ ✰✮ ❙û ❞ö♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ✭→♣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡ ✲ ❑✉tt❛✮ ■♥[1] := f [x❴, y❴] := y − x − 1; x0 = 1; y0 = 2; xf = 2; n = 10; h = (xf − x0)/n; rk[x0❴, xf ❴, y0❴, n❴] := M odule[{h, x, y, X, Y, i, j, k1, k2, k3, k4, k}, h = (xf − x0)/n; x = x0; y = y0; ✷✸ X = {x}; Y = {y}; Do[k1 = f [x, y]; k2 = f [x + h/2, y + h ∗ k1/2]; k3 = f [x + h/2, y + h ∗ k2/2]; k4 = f [x + h, y + h ∗ k3]; k = (k1 + 2k2 + 2k3 + k4)/6; y = y + h ∗ k; x = x + h; X = Append[X, x]; Y = Append[Y, y], {i, 1, n}]; {X, Y }]; X, Y = rk[0., 1., 1, 10]; P rint[T able[T ranspose[{X, Y }]]] Out[1] = {{0., 1}, {0.5, 0.851563}, {1., 0.282654}} ■♥[2] := f [x❴, y❴] := e−x + 1+x ; x0 = 0; y0 = 1; xf = 1; n = 2; h = (xf − x0)/n; rk[x0❴, xf ❴, y0❴, n❴] := M odule[{h, x, y, X, Y, i, j, k1, k2, k3, k4, k}, h = (xf − x0)/n; x = x0; y = y0; X = {x}; Y = {y}; Do[k1 = f [x, y]; k2 = f [x + h/2, y + h ∗ k1/2]; k3 = f [x + h/2, y + h ∗ k2/2]; k4 = f [x + h, y + h ∗ k3]; k = (k1 + 2k2 + 2k3 + k4)/6; ✷✹ y = y + h ∗ k; x = x + h; X = Append[X, x]; Y = Append[Y, y], {i, 1, n}]; {X, Y }]; X, Y = rk[0., 1., 1, 2]; P rint[T able[T ranspose[{X, Y }]]] Out[2] = {{0., 1}, {0.1., 1.194998}, {0.2., 1.37969}, {0.3., 1.5536}, {0.4., 1.71613}, 0.5., 1.8665}, {0.6., 2.00516}, {0.7., 2.13131}, {0.8., 2.24546}, {0.9., 2.3481}, {1., 2.43991}} ✰✮ ❙û ❞ö♥❣ ▼❛t❤❡♠❛t✐❝❛ ✤➸ ✈➩ ỗ t ữỡ tr ✈✐ ♣❤➙♥ ■♥[1] := P lot[x + − ex, {x, 0, 1}]; ❖✉t ❬✶❪❂ ✷✺ ❱➼ ❞ö ✸✳✸ ❍➻♥❤ ỗ t ữỡ tr s ①➨t y˙ = e−x + 1+x ❍➻♥❤ ✸✳✷✿ ỗ t ởt số t ♠➔ ❝❤ó♥❣ t❛ ✤➣ ✤÷đ❝ ❤å❝ ✈➔ t❤÷í♥❣ ①✉②➯♥ →♣ ❞ư♥❣ tù❝ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ t❤õ ❝ỉ♥❣ ❜➡♥❣ ❝→❝❤ ❦❤↔♦ st t tr q ữợ t ①→❝ ✤à♥❤✳ ✰ ❑❤↔♦ s→t sü ❜✐➳♥ t❤✐➯♥ ✭tù❝ ❧➔ t ỗ tr t ởt số ỗ t❤à ❤➔♠ sè ✤✐ q✉❛✱ s❛✉ ✤â ❞ü❛ ✈➔♦ sü ❜✐➳♥ t❤✐➯♥ ✤➸ ♥è✐ ❝→❝ ✤✐➸♠ ✤â ❧↕✐ ✈ỵ✐ ♥❤❛✉ t ữủ ỗ t số ỗ t ❤➔♠ sè s➩ ❜✐➸✉ ❞✐➵♥ sü ♣❤ö t❤✉ë❝ ❝õ❛ ❤➔♠ ② ✈➔♦ ❜✐➳♥ ①✳ ●✐❛♦ ✤✐➸♠ ❝õ❛ trö❝ t✉♥❣ ✈➔ trö❝ ❤♦➔♥❤ ❧➔ t↕✐ ❣è❝ tå❛ ✤ë ❖✳ ❱➻ ✈➟② ữủ ỗ t ởt số tố ❦❤→ ♥❤✐➲✉ t❤í✐ ❣✐❛♥✱ ✤ỉ✐ ❦❤✐ ❝á♥ ❣➦♣ ❦❤â ❦❤➠♥ ❦❤✐ ♠ët ❤➔♠ sè r➜t ❦❤â ✤➸ ❦❤↔♦ s→t sü ❜✐➳♥ t❤✐➯♥ ♥❤÷✿ ❍➔♠ ♠ơ✱ ❤➔♠ ❧♦❣❛r✐t✱ ❤➔♠ ❧÷đ♥❣ ❣✐→❝✳✳✳ tr ởt số trữớ ủ t t ỗ t❤à ✤✐ q✉❛ ❣✐ú❛ ① ✈➔ ② ❝→❝❤ ①❛ ♥❤❛✉ ❞➝♥ ✤➳♥ sü ❝❤➯♥❤ ❧➺❝❤ ❦❤→ ❧ỵ♥ ❣✐ú❛ trư❝ t✉♥❣ trử ỗ t r õ ❦✐➸♠ s♦→t tr♦♥❣ tr❛♥❣ ❣✐➜②✳ ❚r♦♥❣ ❦❤✐ ✤â ✈✐➺❝ sû tt ợ t ỗ t❤à ❤➔♠ sè ✤➣ ✤❡♠ ❧↕✐ ♥❤✐➲✉ ÷✉ ✤✐➸♠ ✈÷đt trë✐ ❤ì♥ ✤â ❧➔✿ ✰ ❚è♥ ➼t t❤í✐ ❣✐❛♥ ❝❤➾ ❞ò♥❣ ♠ët ❝➙✉ ❧➺♥❤✿ P❧♦t❬❢✭①✮✱①✱①♠✐♥✱①♠❛①❪ ✰ ❱➩ ❜➡♥❣ ♠→② ♥➯♥ ỗ t số ỡ ❣✐ú❛ ❝→❝ ❦❤♦↔♥❣ tr➯♥ ♠ët trö❝ ❧➔ ❜➡♥❣ ♥❤❛✉ ✈➻ t s tỹ ữợ ữủ trử ỗ t ỡ ố ❤ì♥✳ ✰ ❈❤ó♥❣ t❛ ❝â t❤➸ ✈➩ ❝ị♥❣ ❧ó❝ ♥❤✐➲✉ ❤➔♠ sè tr➯♥ ❝ị♥❣ ♠ët ❤➺ trư❝ tå❛ ✤ë ♥❤÷♥❣ ❝ơ♥❣ ❝❤➾ ❝➛♥ ❞ị♥❣ ♠ët ❝➙✉ ❧➺♥❤✳ ❱➼ ❞ư✿ P❧♦t❬❢✭①✮✱❣✭①✮✱①✱①♠✐♥✱①♠❛①❪ r trữớ ủ ỗ t sè t❤➻ ❝❤ó♥❣ t❛ ❝â t❤➸ t➠♥❣ ❤♦➦❝ ❣✐↔♠ ❦❤♦↔♥❣ ợ tr trử t ữủ sỹ tữỡ ỳ ỗ t õ ữ ỗ t tt ổ tr q ữợ tổ tữớ t s tỹ ỷ ỵ r ởt ỗ t ❤♦➔♥ ❝❤➾♥❤✱ ✈➻ ✈➟② ♠➔ q✉→ tr➻♥❤ ✈➩ s➩ ♥❤❛♥❤ ❤ì♥ ✈➔ ✈➩ ✤÷đ❝ t➜t ❝↔ ❝→❝ ❤➔♠ sè ♠➔ ❦❤ỉ♥❣ ❣➦♣ ❦❤â ❦❤➠♥ ❣➻ ♥❤÷ ❜➡♥❣ ❝→❝❤ ✈➩ t❤õ ổ r ỳ trữớ ủ ũ ỗ t ✤à♥❤ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t❤➻ ✈✐➺❝ sû ❞ư♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ❧↕✐ ❝➔♥❣ ❝➛♥ t❤✐➳t ✈➔ t✐➺♥ ❧ñ✐ ỡ õ t ỗ t số ❝õ❛ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ❝â ♥❤✐➲✉ ÷✉ ✤✐➸♠✿ ♥❤❛♥❤✱ ✤➭♣✱ ❝❤➼♥❤ ①→❝✱ t✉② ♥❤✐➯♥ tr♦♥❣ ♠ët sè ❜➔✐ t♦→♥ ❝ô♥❣ ♣❤↔✐ ✈➩ t❤❡♦ ❝→❝❤ t❤ỉ♥❣ t❤÷í♥❣✳ ✷✼ ✐✐✮ ❙û ❞ư♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ✤➸ ❣✐↔✐ ♠ët sè ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ s❛✉✿ +) y˙ = (1/4)(1 + y ); y(0) = 1 ■♥[1] := DSolve[{y[x] ˙ == (1 + y[x]y[x]), y[0] == 1}, y, x] Out[1] = {{y → F unction[{x}, T an[ Π+x ]]}} +) y˙ = 2xy ; y(0) = ■♥[2] := DSolve[{y[x] ˙ == 2xy[x]y[x], y[0] == 1}, y, x] Out[2] = {{y → F unction[{x}, ]}} − x2 +) xy˙ = 3x − 2y; y(2) = ■♥[3] := DSolve[{y[x] ˙ == 3x − 2y[x], y[2] == 3}, y, x] + x3 Out[3] = {{y → F unction[{x}, ]}} x2 +) y˙ + 2y = 4x ■♥[4] := DSolve[{y[x] ˙ + 2y[x] == 4x, y, x] Out[4] = {{y → F unction[{x}, 4( −1 x + ) + e−2x C[1]]}} +) y˙ + y = xy −1 ■♥[5] := DSolve[{y[x] ˙ + y[x] == x , y, x] y[x] Out[5] = {{y → F unction[{x}, − 2( = {{y → F unction[{x}, −1 x + ) + e−2x C[1]]}} x −2x C[1]]}} 2( −1 + 2) + e ✸✳✷✳✷ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ❤❛✐ ❱➼ ❞ö ✸✳✹ ✐✮ ●✐↔✐ ữỡ tr yă 4y + 4y = 2ex ữỡ tổ tữớ yă 4y + 4y = 0✳ k − 4k + = 0, ⇔ k = 2✳ ❚❛ ❝â ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t ❧➔✿ P❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣✿ ◆❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t ❧➔✿ y = C1 e2x + C2 xe2x ✷✽ ◆❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❝â ❞↕♥❣✿ y = C1 (x)e2x + C2 (x)xe2x ✳ ❱ỵ✐✿ C˙ e2x + C˙ e2x = 0, C˙ (e2x ) + C˙ (xe2x ) = 2ex ⇔ C˙ e2x + C˙ e2x = 0, 2C˙ (e2x ) + C˙ (e2x + 2xe2x ) = 2ex ⇔ C˙ + xC˙ = 0, 2C˙ + C˙ (1 + 2x) = 2e−x C˙ = −xC˙ , C˙ = 2e−x ⇔ ⇔ ⇔ C˙ = −2xe−x , C˙ = 2e−x C1 = 2xe−x + 2e−x + A, C2 = −2e−x + B ❱➟② ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❧➔✿ y = (2xe−x + 2e−x + A)e2x + (−2e−x + B)xe2x = = 2ex + Ae2x + Bxe2x ✰✮●✐↔✐ ❜➡♥❣ ▼❛t❤❡♠❛t✐❝❛✿ In[1] := DSolve[ă y [x] 4y[x] + 4y[x] == 2ex , y, x] Out[1] = {{→ F unction[{x}, e2x C1 + e2x xC2 + 2ex ]}} (−2 + log[e])2 ✐✐✮ ❙û ❞ư♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ❣✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ s yă 2y + y = 6xex In[1] := DSolve[ă y [x] − 2y[x] ˙ + y[x] == 6xex , y, x] Out[1] = {{y → F unction[{x}, ex C1 + ex xC2 + ✰✮ 6ex (−2 − x + xlog[e]) ]}} (1 + log[e])3 yă + y = In[2] := DSolve[ă y [x] + y[x] == 3, y, x] Out[2] = {{y → F unction[{x}, 3x − e−x C[1] + C[2]]}} yă + y = 2sinx In[3] := DSolve[ă y [x] + y[x] == 2sinx, y, x] Out[3] = {{y → F unction[{x}, 2sin[x] + C[1]cos[x] + C[2]sin[x]]}} ❝✳ ◆❤➟♥ ①➨t ❚ø ❝→❝ ♠ö❝ ✭❛✮ ✈➔ ✭❜✮ t❛ ♥❤➟♥ t❤➜② r➡♥❣✿ ✣➸ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ❜➟❝ ❤❛✐ t❤➻ q✉→ tr➻♥❤ ❣✐↔✐ ♣❤↔✐ tè♥ ♥❤✐➲✉ t❤í✐ ❣✐❛♥✱ ♣❤↔✐ t❤ü❝ ❤✐➺♥ ❧➛♥ ❧÷đt tứ ữợ r q tr ổ ♣❤↔✐ ❦❤â ❦❤➠♥ tr♦♥❣ ✈✐➺❝ t➼♥❤ ❝→❝ t➼❝❤ ♣❤➙♥✱ ✈✐➺❝ ①→❝ ✤à♥❤ ❝→❝ ❤➡♥❣ sè ❈ ❦❤ỉ♥❣ ✤ì♥ ❣✐↔♥ ✤è✐ ợ ữỡ tr t t số ❦❤ỉ♥❣ ✤ê✐✳ ◆❣♦➔✐ r❛ ♠ët sè ♣❤÷ì♥❣ tr➻♥❤ ❝❤♦ r❛ ♥❣❤✐➺♠ ♥❤÷♥❣ ❦❤ỉ♥❣ t❤➸ t➻♠ ✤÷đ❝ ❝→❝ ❣✐→ trà ❝❤➼♥❤ ①→❝ ❝õ❛ ② t↕✐ ❝→❝ ✤✐➸♠ ①✳ ❱➻ ✈➟② ❝➛♥ ♣❤↔✐ sû ❞ư♥❣ ✤➳♥ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ❣➛♥ ✤ó♥❣ ♠➔ ❝ư t❤➸ ð ✤➙② ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡ ✲ ❑✉tt❛✱ ❞ị♥❣ ♣❤÷ì♥❣ ♣❤→♣ ♥➔② ♣❤↔✐ tr↔✐ q✉❛ ♥❤✐➲✉ ữợ ợ ố ữủ t t ợ ữ ❝ơ♥❣ ❝❤➾ ❝❤♦ ❣✐→ trà ❣➛♥ ✤ó♥❣✳ ❚r♦♥❣ ❦❤✐ ✤â ✈✐➺❝ sû ❞ư♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ✤➸ ❣✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ✤➣ ✤❡♠ ❧↕✐ ❤✐➺✉ q✉↔ ✤→♥❣ ❦➸ ✈ỵ✐ ÷✉ ✤✐➸♠ ❧➔✿ ♥❤❛♥❤✱ ❝❤➾ ❝➛♥ ♥❤➟♣ ❝❤÷ì♥❣ tr➻♥❤ ✤ó♥❣ ❝➜✉ tró❝ s➩ ❝❤♦ ❦➳t q✉↔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ♥❣❛②✱ tø ✤â ❝â t❤➳ t➼♥❤ ❝→❝ ❣✐→ trà ❝❤➼♥❤ ①→❝ ❝õ❛ ❤➔♠ ✸✵ ② t↕✐ ❝→❝ ✤✐➸♠ ①✳ ❚ø ❝→❝ ✈➼ ❞ö ❝→❝ ♠ö❝ tr➯♥ t❛ t❤➜② r➡♥❣ sû ❞ư♥❣ ▼❛t❤❡♠❛t✐❝❛ ♥❤❛♥❤ ❤ì♥ ❣➜♣ ♥❤✐➲✉ ❧➛♥ s♦ ✈ỵ✐ ❣✐↔✐ ❜➡♥❣ t❛② t❤ỉ♥❣ t❤÷í♥❣ ❜➡♥❣ ❝→❝❤ →♣ ❞ư♥❣ ❝→❝ ♣❤÷ì♥❣ ố ợ ởt số ữỡ tr ❝❛♦ t❤➻ r➜t ❦❤â ❦❤✐ ❣✐↔✐ ❜➡♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ tổ tữớ ữ ợ tt t ổ õ ❣➻✳ ◆❤÷♥❣ ♥❣♦➔✐ ♥❤ú♥❣ ÷✉ ✤✐➸♠ ✈÷đt trë✐ ✤â t❤➻ tr♦♥❣ q✉→ tr➻♥❤ sû ❞ö♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ❣➦♣ ♠ët sè ❦❤â ❦❤➠♥ ❧➔✿ tr♦♥❣ ♠ët sè ➼t tr÷í♥❣ ❤đ♣ ♥❣❤✐➺♠ r❛ r➜t ♣❤ù❝ t↕♣✱ ❦❤â ❤✐➸✉✱ ♠ët sè ❝→❝❤ ❜✐➸✉ ❞✐➵♥ ❝õ❛ ❤➔♠ ♠→② t➼♥❤ ❦❤æ♥❣ t❤➸ ❤✐➸✉ ♥➯♥ ♣❤↔✐ ❜✐➸✉ t❤à ❞➔✐ ❞á♥❣✳ ❚r♦♥❣ ❝ò♥❣ ♠ët ❯♥✐t✐t❧❡❞ ❝â t❤➸ ♥❤➟♥ trò♥❣ ❣✐→ trà ❞➝♥ ✤➳♥ ❦➳t q✉↔ s❛✐ ❤♦➦❝ ❜→♦ ❧é✐✳ ◗✉❛ ✤â t❛ t❤➜② r➡♥❣ t✉② ▼❛t❤❡♠❛t✐❝❛ ỏ tỗ t ởt ữủ ữ ự ❞ư♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ✤➸ ❣✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✤➣ ♠❛♥❣ ❧↕✐ ❝❤♦ ♥❣÷í✐ sû ❞ư♥❣ ❤✐➺✉ q✉↔ ✤→♥❣ ❦➸✱ t✐➳t ❦✐➺♠ ✤÷đ❝ t❤í✐ ❣✐❛♥✱ ❝❤♦ ✤ë ❝❤➼♥❤ ①→❝ ❝❛♦✳ ✣➦❝ ❜✐➺t tr♦♥❣ ♥❤ú♥❣ tr÷í♥❣ ❤đ♣ ♠➔ ❝❤ó♥❣ t❛ ❝❤➾ ❝➛♥ t➻♠ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➸ sû ❞ö♥❣ ✈➔♦ ❝→❝ ❜➔✐ t♦→♥ ❦❤→❝ t❤➻ ❣✐↔✐ ❜➡♥❣ ▼❛t❤❡♠❛t✐❝❛ ❧↕✐ ❝➔♥❣ ❝➛♥ t❤✐➳t✳ ❱➔ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝❤➾ ❧➔ ♠ët ❝❤ù❝ ♥➠♥❣ ♥❤ä ❝õ❛ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛✳ ▼❛t❤❡♠❛t✐❝❛ ❝á♥ ❝❤♦ r➜t ♥❤✐➲✉ ù♥❣ ❞ö♥❣ tr♦♥❣ ❝→❝ ♠➦t ❦❤→❝ ❝õ❛ ♥❣➔♥❤ t♦→♥ ❤å❝✳ ✸✶ ❱⑨■ ◆➆❚ ❱➋ ❈❆❘▲ ❘❯◆●❊ ❱⑨ ❲■▲❍❊▲▼ ❑❯❚❚❆ ✯✮ ❈❛r❧ ❘✉♥❣❡ ✭✶✽✺✼✲✶✾✷✼✮ ❧➔ ♥❣÷í✐ ❝♦♥ tr❛✐ t❤ù ❜❛ tr♦♥❣ ❜è♥ ♥❣÷í✐ ❝♦♥ ❝õ❛ ♠ët ❣✐❛ ✤➻♥❤ t❤÷ì♥❣ ❣✐❛ ♥❣÷í✐ ✣ù❝✳ ❈↔ ❜❛ ❛♥❤ ❡♠ ✤➲✉ t❤❡♦ ✤✉ê✐ sü ♥❣❤✐➺♣ t❤÷ì♥❣ ♠↕✐✱ ❜✉ỉ♥ ❜→♥✳ ❘✐➯♥❣ ỉ♥❣ ✤➣ ❝❤å♥ ❝♦♥ ✤÷í♥❣ ❤å❝ t❤ù❝✳ ✶✾ t✉ê✐ ỉ♥❣ rí✐ tr÷í♥❣ ✣↕✐ ❤å❝ ✈➔ ❝❤➾ s❛✉ ♠ët t❤í✐ ❣✐❛♥ ♥❣➢♥ ỉ♥❣ ✤➣ t❤❡♦ ❤å❝ t↕✐ tr÷í♥❣ ✣↕✐ ❤å❝ ▼✉♥✐❝❤ ð ✣ù❝ ✤➸ ♥❣❤✐➯♥ ❝ù✉ ❱➠♥ ❤å❝✳ ◆❤÷♥❣ ❝❤➾ s❛✉ ✻ t✉➛♥ ❝õ❛ ❦❤â❛ ❤å❝ æ♥❣ ✤➣ t❤❛② q ự t ỵ ❤å❝✱ s❛✉ ✤â æ♥❣ ❝❤➾ t➟♣ tr✉♥❣ ♥❣❤✐➯♥ ❝ù✉ ✈➔♦ ❚♦→♥ ❤å❝✳ ➷♥❣ ✤➣ ♥❤➟♥ ❤å❝ ✈à ❚✐➳♥ s➽ t↕✐ tr÷í♥❣ ✣↕✐ ❤å❝ ▼✉♥✐❝❤✳ P❤÷ì♥❣ ♣❤→♣ sè ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ✤↕✐ sè ❧➔ ❤❛✐ ❧➽♥❤ ✈ü❝ ♠➔ æ♥❣ t❤❡♦ ✤✉ê✐ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ✈➲ s❛✉✳ ◆➠♠ ✶✾✵✹ ỉ♥❣ ✤÷đ❝ ❜ê ♥❤✐➺♠ ❧➔♠ ❝❤õ tà❝❤ t♦→♥ ❤å❝ ✈➔ æ♥❣ ✤➣ ❣✐ú ự õ tợ ữ ➷♥❣ ✤➣ ♠➜t s❛✉ ✤â ✷ ♥➠♠✳ ✯✮ ❲✐❧❤❡❧♠ ❑✉tt❛ ✭✶✽✻✼✲✶✾✹✹✮✱ ỉ♥❣ s✐♥❤ r❛ ð P✐ts❝❤❡♥✲t❤÷đ♥❣ ❙✐❧❡❛s✐❛✭ ♥❛② t❤✉ë❝ ❇②❝③②♥❛✲ ❇❛ ▲❛♥✮✳ ✭✶✽✽✺✲✶✽✾✵✮ ỉ♥❣ t❤❡♦ ❤å❝ t↕✐ tr÷í♥❣ ✣↕✐ ❤å❝ ❇r❡s❧❛♥ ✈➔ t✐➳♣ tö❝ ♥❣❤✐➯♥ ❝ù✉ t↕✐ ▼✉♥✐❝❤ ❝❤♦ ✤➳♥ ỡ ổ tr t trủ ỵ r t ✈♦♥ ❲❛❧t❤❡r ❉②❝❦✳ ❚ø ♥➠♠ ✶✽✾✽✱ æ♥❣ ✤➣ ❞➔♥❤ ♠ët ♥➠♠ t↕✐ tr÷í♥❣ ✣↕✐ ❤å❝ ❈❛♠❜r✐❞❣❡✳ ◆➠♠ ✶✾✶✷ ỉ♥❣ trð t❤➔♥❤ ❣✐→♦ s÷ t↕✐ ❙t✉tt❣❛rt✱ ♥ì✐ ỉ♥❣ ð ❧↕✐ ❝❤♦ ✤➳♥ ❦❤✐ ♥❣❤➾ ❤÷✉ ✈➔♦ ♥➠♠ ✶✾✺✸✳ ❑✉tt❛ ✤➣ ♠➜t ✈➔♦ ♥➠♠ ✶✾✹✹✳ ◆➠♠ ✶✾✵✶ ❈❛r❧ ❘✉♥❣❡ ✈➔ ❲✐❧❤❡❧♠ ❑✉tt❛ ✤➣ ❝ị♥❣ ♥❤❛✉ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ♣❤→t tr✐➸♥ ♣❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡ ✲ ❑✉tt❛✳ P❤÷ì♥❣ ♣❤→♣ ♥➔② ❞ị♥❣ ✤➸ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✳ ✸✷ ❑➌❚ ▲❯❾◆ ✣➲ t➔✐ ✑ Ù♥❣ ❞ư♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ❝❤♦ ♣❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡ ✲ ❑✉tt❛ ✑ ✤➣ t❤ü❝ ❤✐➺♥ ✤÷đ❝ ♥❤ú♥❣ ♥ë✐ ❞✉♥❣ s❛✉✿ ❛✳ ❑❤→✐ q✉→t ❧↕✐ ❦✐➳♥ t❤ù❝ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❜✳ ❳➙② ❞ü♥❣ ❝æ♥❣ t❤ù❝ ①➜♣ ①➾ ❜➟❝ ❜è♥ ❘❑ ✤➸ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❜➟❝ ♠ët✳ ❝✳ ▼ð rë♥❣ ❝ỉ♥❣ t❤ù❝ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ♥✱ ✭♥❃✶✮ ♠➔ ❝ư t❤➸ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ✷ ✈➔ ❤➺ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❜➟❝ ♥❤➜t✳ ❞✳ ●✐ỵ✐ t❤✐➺✉ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ✲ ♣❤➛♥ ♠➲♠ ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ tr♦♥❣ t➼♥❤ t♦→♥ t♦→♥ ❤å❝✳ ❚r♦♥❣ ✣➲ t➔✐ ✤➣ ✤➲ ❝➟♣ ❝❤ù❝ ♥➠♥❣ ữỡ tr ỗ t ❞✐➵♥ ♥❣❤✐➺♠ ✧P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✧ ❝õ❛ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛✳ ◗✉❛ ✤â s♦ s→♥❤ ✤÷đ❝ ÷✉ ✤✐➸♠ ✈➔ ♥❤÷đ❝ ✤✐➸♠ ❝õ❛ ✈✐➺❝ ❣✐↔✐ ❜➡♥❣ t❛② ✈ỵ✐ ✈✐➺❝ →♣ ❞ư♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✈➔ ✈✐➺❝ ❣✐↔✐ ❜➡♥❣ ♠→② t➼♥❤ ❦❤✐ sû ❞ư♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛✳ ❉♦ t❤í✐ ❣✐❛♥ ♥❣➢♥ ✈➔ ❦✐➳♥ t❤ù❝ ❝á♥ ♥❤✐➲✉ ❤↕♥ ❝❤➳ ♥➯♥ ▲✉➟♥ ✈➠♥ ❝❤ó♥❣ ❡♠ ❝❤➾ ❞ø♥❣ ❧↕✐ t↕✐ ✤➙②✳ ❍② ✈å♥❣ ▲✉➟♥ ✈➠♥ ❝á♥ ✤÷đ❝ t✐➳♣ tư❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥ ✤➸ ❤✐➸✉ rã ❤ì♥ ❝→❝ ù♥❣ ❞ư♥❣ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❘✉♥❣❡ ✲ ❑✉tt❛✱ ✈➔ s➩ trð t❤➔♥❤ ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❤✐➺✉ q✉↔ ♥❤➜t✳ ✸✸ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❊❞✇❛r❞s ❈✳❍❡♥❞r②✱ ❉❛✈✐❞ ❊ P❡♥♥❡②✳ ❊❧❡♠❡❡♥t❛r② ❞✐❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✇✐t❤ ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠✳ Pr❡♥t✐❝❡ ❍❛❧❧✱ ✷✵✵✼✱ ✼✾✷♣ ✳ ❬✷❪ P❤↕♠ ❑ý ❆♥❤ ●✐↔✐ t➼❝❤ sè✳ ◆❤➔ ①✉➜t ❜↔♥ ❍➔ ◆ë✐✳ ✳ ❬✸❪ ❚r➛♥ ❆♥❤ ❇↔♦✱ ◆❣✉②➵♥ ❱➠♥ ❑❤↔✐✱ P❤↕♠ ❱➠♥ ❑✐➲✉✱ ◆❣ỉ ❳✉➙♥ ❙ì♥✳ ●✐↔✐ t➼❝❤ sè✳ ◆❤➔ ①✉➜t ❜↔♥ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠✲✷✵✵✼✳ ❬✹❪ ◆❣✉②➵♥ ❚❤➳ ❍♦➔♥ ✲ P❤↕♠ P❤✉✳ ❈ì sð ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✈➔ ỵ tt t ❬✺❪ ◆❣✉②➵♥ ✣➻♥❤ ❚r➼ ✭❝❤õ ❜✐➯♥✮ ✲ ❚↕ ❱➠♥ ✣➽♥❤ ỗ ý t P❤➨♣ t➼♥❤ ❣✐↔✐ t➼❝❤ ♥❤✐➲✉ ❜✐➳♥ sè✳ ◆❤➔ ①✉➜t ❜↔♥ ●✐→♦ ❞ö❝ ✸✹