Phương pháp xấp xỉ euler trong phương trình vi phân thường

❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖ ✣❸■ ❍➴❈ ✣⑨ ◆➂◆● ✖✖✖✖✖ ◆●❯❨➍◆ ❚❍➚ ◆❍× ❆◆❍ P❍×❒◆● P❍⑩P ❳❻P ❳➓ ❊❯▲❊❘ ❚❘❖◆● P❍×❒◆● ❚❘➐◆❍ ❱■ P❍❹◆ ❚❍×❮◆● ▲❯❾◆ ❱❿◆ ❚➮❚ ◆●❍■➏P ữợ ▼ö❝ ❧ö❝ ▼Ð ✣❺❯ ✶ ▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ✶✳✶ ✶✳✷ ✸ ✺ P❍×❒◆● ❚❘➐◆❍ ❱■ P❍❹◆ ❚❍×❮◆● ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛ ✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✶✳✷ ◆❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ✺ ✳ ✳ ✳ ✳ ✳ ❙Ü ❚➬◆ ❚❸■ ❱⑨ ❉❯❨ ◆❍❻❚ ◆●❍■➏▼ ❈Õ❆ P❍×❒◆● ❚❘➐◆❍ ❱■ P❍❹◆ ❚❍×❮◆● ❇❾❈ ◆❍❻❚ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷✳✶ ✣à♥❤ ♥❣❤➽❛ ✈➲ ✤✐➲✉ ❦✐➺♥ ▲✐♣s❝❤✐t③ ✳ ✳ ✳ ✳ ✳ ✳ ỵ sỹ tỗ t t ữỡ tr ♣❤➙♥ t❤÷í♥❣ ❜➟❝ ♥❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷ P❍×❒◆● P❍⑩P ❊❯▲❊❘ ❈❍❖ P❍×❒◆● ❚❘➐◆❍ ❱■ P❍❹◆ ❚❍×❮◆● ✼ ✷✳✶ ✷✳✷ ✷✳✸ ●■❰■ ❚❍■➏❯ P❍×❒◆● P❍⑩P ❊❯▲❊❘ ✳ ✳ ✳ ✳ ỗ ố ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✶✳✷ ❚❤➔♥❤ ❧➟♣ ❝æ♥❣ t❤ù❝ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r ✳ ✳ ✳ ✳ ✽ ✷✳✶✳✸ ❚❤✉➟t t♦→♥ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❙❆■ ❙➮ ❚❘❖◆● P❍×❒◆● P❍⑩P ❊❯▲❊❘ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✷✳✶ ❙❛✐ sè ✤à❛ ♣❤÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✷✳✷ ❙❛✐ sè t➼❝❤ ❧ô②✭s❛✐ sè t♦➔♥ ♣❤➛♥✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ P❍×❒◆● P❍⑩P ❊❯▲❊❘ ❚❘❖◆● P❍×❒◆● ❚❘➐◆❍ ❱■ P❍❹◆ ❚❍×❮◆● ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹ ✶✵ ✶✽ ▼❐❚ ❙➮ ❇⑨■ ❚❖⑩◆ ❚➐▼ ◆●❍■➏▼ ●❺◆ ✣Ĩ◆● ❱❰■ P❍×❒◆● P❍⑩P ❳❻P ❳➓ ❊❯▲❊❘ ❚❘❖◆● P❍×❒◆● ❚❘➐◆❍ ❱■ P❍❹◆ ❚❍×❮◆● ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✾ ✷✳✺ ×❯ ✣■➎▼✲ ❚➑◆❍ ✃◆ ✣➚◆❍ ❈Õ❆ P❍×❒◆● P❍⑩P ❳❻P ❳➓ ❊❯▲❊❘ ❚❘❖◆● P❍×❒◆● ❚❘➐◆❍ ❱■ P❍❹◆ ❚❍×❮◆● ✷✷ ✷✳✻ ❍❸◆ ❈❍➌ ❚❘❖◆● P❍×❒◆● P❍⑩P ❊❯▲❊❘ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✼ P❍⑩❚ ❚❘■➎◆ ❱⑨ ❙❖ ❙⑩◆❍ P❍×❒◆● P❍⑩P ❊❯▲❊❘ ❱❰■ ▼❐❚ ❙➮ P❍×❒◆● P❍⑩P ❑❍⑩❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✺ ✸ Ù◆● ❉Ö◆● P❍❺◆ ▼➋▼ ▼❆❚❍❊▼❆❚■❈❆ ❈❍❖ P❍×❒◆● P❍⑩P ❳❻P ❳➓ ❊❯▲❊❘ ✷✽ ✸✳✶ ●■❰■ ❚❍■➏❯ ❱➋ P❍❺◆ ▼➋▼ ▼❆❚❍❊▼❆❚■❈❆ ✳ ✳ ✳ ✳ ✳ ✸✳✷ Ù◆● ❉Ư◆● P❍❺◆ ▼➋▼ ▼❆❚❍❊▼❆❚■❈❆ ❈❍❖ P❍×❒◆● P❍⑩P ❳❻P ❳➓ ❊❯▲❊❘ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❑➌❚ ▲❯❾◆ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✷✽ ✷✾ ✸✸ ✸✹ ✸ é ỵ ỹ t r ❦❤♦❛ ❤å❝ ❦ÿ t❤✉➟t ❝❤ó♥❣ t❛ t❤÷í♥❣ ❣➦♣ r➜t ♥❤✐➲✉ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣✱ t✉② ♥❤✐➯♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② t❤÷í♥❣ ♣❤ù❝ t↕♣ ♠➔ tr♦♥❣ ♠ët sè tr÷í♥❣ ❤đ♣ ❝ơ♥❣ ❦❤ỉ♥❣ t❤➸ t➻♠ ✤÷đ❝ ♥❣❤✐➺♠ t÷í♥❣ ♠✐♥❤✳ ❍ì♥ ♥ú❛✱ ✈➻ ❝→❝ ❝ỉ♥❣ t❤ù❝ ♥❣❤✐➺♠ tữớ ự t ỗ st t➼♥❤ ❝❤➜t ❝õ❛ ♥â ❝á♥ ❣➦♣ ♥❤✐➲✉ ❦❤â ❦❤➠♥✳ ❚r♦♥❣ ❦ÿ t❤✉➟t✱ ♥❣÷í✐ t❛ sû ❞ư♥❣ ❝→❝ ❣✐→ trà t❤✉ ✤÷đ❝ ❜➡♥❣ ✈✐➺❝ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ❝→❝ ✭❤➺✮ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣✳ ❇ð✐ ✈➟②✱ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ✤➸ t➻♠ ♥❣❤✐➺♠ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ♥➔② trð ♥➯♥ ❝➜♣ t❤✐➳t ✈➔ tü ♥❤✐➯♥✳ Ð t❤➳ ❦➾ ❳❱■■■✱ ❝→❝ ✈➜♥ ✤➲ t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ✤÷đ❝ ▲♦❡♥❤❛r❞ ❊✉❧❡r✲ ♥❤➔ t♦→♥ ❤å❝ ♥❣÷í✐ ❚❤ư② ❙➽ ♣❤→t tr✐➸♥ ✈➔ t❤✉ ✤÷đ❝ ♥❤✐➲✉ t❤➔♥❤ tü✉ rü❝ rï✳ t ổ ữủ t ợ s t↕♦ r❛ ♠ët ❝❤✉é✐ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ t➼♥❤ ①➜♣ ①➾✱ ✤÷đ❝ sû ❞ư♥❣ ♥❤✐➲✉ tr♦♥❣ t➼♥❤ t♦→♥✱ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ♥ê✐ t✐➳♥❣ ♥❤➜t tr♦♥❣ ✤â ❝❤➼♥❤ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r✳ ❱ỵ✐ ♠♦♥❣ ♠✉è♥ ❝â t❤➸ ❤✐➸✉ ❦➽ ❤ì♥ ✈➲ ❝→❝ ❞↕♥❣ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ữủ sỹ ủ ỵ ữợ ✕ ❚❙✳ ▲➯ ❍↔✐ ❚r✉♥❣ ♥➯♥ ❝❤ó♥❣ tỉ✐ ❧ü❛ ❝❤å♥ ✤➲ t➔✐ ✿ ✓P❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ❊✉❧❡r tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣✔ ❝❤♦ ❧✉➟♥ ✈➠♥ tèt ♥❣❤✐➺♣ ❝õ❛ ✷✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ✤➲ t➔✐ ♠➻♥❤✳ ▼ö❝ ✤➼❝❤ ❝õ❛ ✤➲ t➔✐ ❧➔ sû ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ❊✉❧❡r ✤➸ ①❡♠ ①➨t ✈➔ t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣✱ tø ✤â s♦ s→♥❤ s số ợ ữỡ tr õ ỗ tớ ự ự ▼❛t❤❡♠❛t✐❝❛ ✤➸ ✈✐➳t ♣❤÷ì♥❣ tr➻♥❤ t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ t❤❡♦ ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ❊✉❧❡r ✈➔ ♠ỉ t↔ ♥❣❤✐➺♠ ❝❤➼♥❤ ữỡ tr tữớ ỗ t❤à t❤æ♥❣ q✉❛ ❝→❝ ❣â✐ ❝➙✉ ❧➺♥❤ ✤➣ ✸✳ ✣è✐ t÷đ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ✤➲ t➔✐ ✤÷đ❝ ❧➟♣ tr➻♥❤✳ ✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉✿ ◆❣❤✐➯♥ ❝ù✉ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r ✤➸ t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ✈➔ ❧➟♣ tr➻♥❤ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r tr♦♥❣ ▼❛t❤❡♠❛t✐❝❛✳ P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉✿ ◆❣❤✐➯♥ ❝ù✉ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r ❝❤♦ ❝→❝ ❜➔✐ t♦→♥ ✹✳ Þ ♥❣❤➽❛ ❦❤♦❛ ❤å❝ ✈➔ t❤ü❝ t t tr ỵ tt ữỡ tr tữớ t õ ỵ t ỵ t❤✉②➳t✱ ❝â t❤➸ sû ❞ư♥❣ ♥❤÷ ❧➔ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❞➔♥❤ ❝❤♦ s✐♥❤ ✈✐➯♥ ✈➔ ❝→❝ ✤è✐ t÷đ♥❣ ❝â ♠è✐ q✉❛♥ t➙♠ ✤➳♥ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r ❝❤♦ ♣❤÷ì♥❣ ✺✳ ❈➜✉ tró❝ ❧✉➟♥ ✈➠♥ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣✳ ◆❣♦➔✐ ♣❤➛♥ t ỗ ữỡ ❈❤÷ì♥❣ ✶✿ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✳ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ởt số ỵ sỹ tỗ t t tr ữỡ tr ♣❤➙♥ t❤÷í♥❣ ✈➔ ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ t➼♥❤ ①➜♣ ①➾ t❤÷í♥❣ ❞ị♥❣✳ ❈❤÷ì♥❣ ✷✿ ❚r➻♥❤ ❜➔② ♥ë✐ ❞✉♥❣ ✈➔ s❛✐ sè ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ❊✉❧❡r tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣❀ ✤÷❛ r❛ ♠ët sè ❜➔✐ t♦→♥ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✤÷đ❝ t➻♠ ♥❣❤✐➺♠ ①➜♣ ①➾ t❤❡♦ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r❀ ÷✉ ✤✐➸♠ ✈➔ ❤↕♥ ❝❤➳ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r❀ ỗ tớ t tr s s ữỡ ợ ởt số ữỡ ữỡ ợ t tt tr➻♥❤ ❜➔② ♥❤ú♥❣ ù♥❣ ❞ö♥❣ ❝õ❛ ♣❤➛♥ ♠➲♠ ♥➔② tr♦♥❣ ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ❊✉❧❡r ✺ ❈❤÷ì♥❣ ✶ ▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ✶✳✶ P❍×❒◆● ❚❘➐◆❍ ❱■ P❍❹◆ ❚❍×❮◆● ✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛ ✿ ▼ët ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ❝â ❞↕♥❣ tê♥❣ q✉→t ✿ F (x, y, y , y , , y (n) ) = ✭✶✳✶✮ x ❧➔ ❜✐➳♥ ✤ë❝ ❧➟♣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✮✱ y ❧➔ ➞♥ ❤➔♠✱ y , y , ., y (n) ✤↕♦ ❤➔♠ ❝õ❛ ❤➔♠ y ✳ tr♦♥❣ ✤â ❧➔ ❝→❝ ❈➜♣ ❝õ❛ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ❧➔ ❝➜♣ ❝❛♦ ♥❤➜t ❝õ❛ ✤↕♦ ❤➔♠ ✭❤❛② ✈✐ ♣❤➙♥✮ t❤ü❝ sü ❝â ♥❣❤✐➺♠ tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤✳ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ❝➜♣ ♥ ❝â ❞↕♥❣ tê♥❣ q✉→t ✿ F (x, y, y , y , , y (n) ) = ✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ♥✳ ❚r♦♥❣ ✤â ✭✶✳✷✮ y = y(x) ❧➔ ❤➔♠ ❝➛♥ t➻♠✳ ✶✳✶✳✷ ◆❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ❍➔♠ y = y(x) ✤÷đ❝ ❣å✐ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✭✶✳✶✮ ♥➳✉ ♥❤÷ tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✭✶✳✶✮ t❤❛② y = y(x), y = y (x), , y (n) = y (n) (x) t❛ ♥❤➟♥ ✤÷đ❝✿ F (x, y, y , y , , y (n) ) = ✻ ✶✳✷ ❙Ü ❚➬◆ ❚❸■ ❱⑨ ❉❯❨ ◆❍❻❚ ◆●❍■➏▼ ❈Õ❆ P❍×❒◆● ❚❘➐◆❍ ❱■ P❍❹◆ ❚❍×❮◆● ❇❾❈ ◆❍❻❚ ✶✳✷✳✶ ✣à♥❤ ♥❣❤➽❛ ✈➲ ✤✐➲✉ ❦✐➺♥ ▲✐♣s❝❤✐t③ f (x, y) t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ▲✐♣s❝❤✐t③ tr♦♥❣ ♠✐➲♥ ❉ ❝õ❛ ♠➦t ♣❤➥♥❣ ❖①② t❤❡♦ ❜✐➳♥ y tỗ t số st s❛♦ ❝❤♦ ✈ỵ✐ ∀(x, y1 ), (x, y2 ) D : ❍➔♠ |f (x, y1 ) − f (x, y2 )| L|y1 y2 | ỵ sỹ tỗ t t ữỡ tr ✈✐ ♣❤➙♥ t❤÷í♥❣ ❜➟❝ ♥❤➜t ❈❤♦ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ❜➟❝ ♥❤➜t y = f (x, y), y(x0 ) = y0 ✈➔ ❤➔♠ f (x, y) ❛✳ ❍➔♠ ❝â ♥❤ú♥❣ t➼♥❤ ❝❤➜t s❛✉ ✤➙②✿ f (x, y) ①→❝ ✤à♥❤ ✈➔ ❧✐➯♥ tö❝ t↕✐ ♠ët ♠✐➲♥ ❉ ♥➔♦ ✤â ❝õ❛ ♠➦t ❖①② ❝â ❝❤ù❛ ✤✐➸♠ ✈ỵ✐ (x0 , y0 )✳ f (x, y) tr♦♥❣ ❤➡♥❣ sè L > 0✿ ❜✳ ❍➔♠ ♠✐➲♥ ❉ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ▲✐♣s❝❤✐t③ t❤❡♦ ❜✐➳♥ |f (x, y1 ) − f (x, y2 )| L|y1 y2 | õ tỗ t ♥❤➜t ♠ët ♥❣❤✐➺♠ ✤♦↕♥ ✭✶✳✸✮ [x0 − h, x0 + h], h > 0✳ ✼ ✈ỵ✐ y ∀(x, y1 ), (x, y2 ) D✳ y = y(x) t❤ä❛ ♠➣♥ ✤✐➲✉ tr ữỡ Pì PP Pì ❱■ P❍❹◆ ❚❍×❮◆● ✷✳✶ ●■❰■ ❚❍■➏❯ P❍×❒◆● P❍⑩P ❊❯▲❊❘ ✷✳✶✳✶ ỗ ố ữỡ r t r ổ ♣❤↔✐ t➜t ❝↔ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❞↕♥❣ tê♥❣ q✉→t dy dx = f (x, y) ✤➲✉ ❝â t❤➸ ữủ ữợ t ữỡ ♣❤→♣ ❣✐↔✐ t➼❝❤✳ ❈❤➥♥❣ ❤↕♥✱ ①➨t ♣❤÷ì♥❣ tr➻♥❤✿ dy = e−x dx ✭✷✳✶✮ ◆❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮ ❧➔ ♠ët ♥❣✉②➯♥ ❤➔♠ ❝õ❛ ❝❤ó♥❣ t❛ ❜✐➳t r➡♥❣ ♠å✐ ♥❣✉②➯♥ ❤➔♠ ❝õ❛ f (x) = e−x e−x ✳ ◆❤÷♥❣ ✤➲✉ ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ❤➔♠ ❝➜♣ ♥❣❤➽❛ ❧➔ ổ t ữủ ữợ ởt t❤ù❝ ❣✐↔✐ t➼❝❤✳ ❱➻ ✈➟②✱ ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✶✮ ❦❤æ♥❣ t❤➸ ữợ ởt sỡ ố ❣➢♥❣ sû ❞ư♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ♥❤÷ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✉é✐ ♥❣✉②➯♥ ✤➸ t➻♠ r❛ ♠ët ❝æ♥❣ t❤ù❝ ♥❣❤✐➺♠ ❞↕♥❣ ❤✐➸♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮ ✤➲✉ t❤➜t ❜↕✐✳ ❱➜♥ ✤➲ tr➯♥ ✤➣ ✤÷đ❝ ❣✐↔✐ q✉②➳t tr♦♥❣ t❤➳ ❦➾ ❳❱■■■ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ♠❛♥❣ t➯♥ ♥❤➔ t♦→♥ ❤å❝ ✈➽ ✤↕✐ ❊✉❧❡r✳ ❇➔✐ t♦→♥ ✤÷đ❝ ❧➟♣ ✤➸ ✈➩ ✤÷í♥❣ ❝♦♥❣ ♥❣❤✐➺♠ tø ✤✐➸♠ ✤➛✉ ♣❤÷ì♥❣ tr➻♥❤ y = f (x, y)✳ (x0 , y0 ) ◗✉→ tr➻♥❤ ♥➔② ✤÷đ❝ t✐➳♥ ❤➔♥❤ ♥❤÷ s❛✉ ã ữỡ tr t t ã (x0 , y0 ) ❞✐ ❝❤✉②➸♥ ♠ët ✤♦↕♥ t↕✐ (x0 , y0 ) s➩ ❝â ✤✐➸♠ (x1 , y1 )✳ ❇ót ✈➩ ❜➢t ✤➛✉ tø ✤✐➸♠ ✤➛✉ ❚↕✐ ✤✐➸♠ (x1 , y1 ) út ữợ ♠ët ✤♦↕♥ ♥❤ä t❤❡♦ ♣❤÷ì♥❣ tr➻♥❤ ❝õ❛ t✐➳♣ t✉②➳♥ t↕✐ • ❚↕✐ ✤✐➸♠ ♥❤ä t❤❡♦ (x2 , y2 )✱t÷ì♥❣ (x1 , y1 ) s➩ ❝â ✤✐➸♠ tü ♥❤÷ ✈➟② s➩ ❝â ✤✐➸♠ (x2 , y2 ) ✳ (x3 , y3 )✳ ❍➻♥❤ ✭✷✳✶✮ ♠æ t↔ ❦➳t q✉↔ ❝õ❛ q✉→ tr➻♥❤ ♥â✐ tr➯♥✿ ✤â ❧➔ ✤÷í♥❣ ❣➜♣ ❦❤ó❝ ♥è✐ ❝→❝ ✤✐➸♠ (x0 , y0 ), (x1 , y1 ), (x2 , y2 )✱✳✳✳✳❚✉② ♥❤✐➯♥ ❣✐↔ sû r➡♥❣ ♠é✐ ✑✤♦↕♥ ♥❤ä✑ ❜ót ✈➩ ❞✐ ❝❤✉②➸♥ ❞å❝ t❤❡♦ ✤♦↕♥ ❞è❝ t✐➳♣ t✉②➳♥ r➜t ♥❤ä✱ ♥❤ä ✤➳♥ ♠ù❝ ❜➡♥❣ ♠➢t t❤÷í♥❣ ❦❤ỉ♥❣ t❤➸ ♥❤➟♥ r❛ ✤÷đ❝✳ ❑❤✐ ✤â ✤÷í♥❣ ❣➜♣ ❦❤ó❝ ✤÷đ❝ ①❡♠ ♥❤÷ ữớ trỡ t ụ ỵ ữỡ r ị ♥❣❤➽❛ ❤➻♥❤ ❤å❝ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r ✷✳✶✳✷ ❚❤➔♥❤ ❧➟♣ ❝ỉ♥❣ t❤ù❝ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r ●✐↔✐ ❜➔✐ t♦→♥ ❈❛✉❝❤②✿ dy dx = f (x, y), y(x0 ) = y0 ✭✷✳✷✮ ❱➜♥ ✤➲ ✤÷đ❝ ✤➦t r❛ ❝õ❛ ❜➔✐ t♦→♥ ❧➔ t➻♠ ❣➛♥ ✤ó♥❣ ❤➔♠ ♥❣❤✐➺♠ ②✭①✮ t↕✐ x1 , x2 , x3 , ✱tù❝ ❧➔ t➼♥❤ ❝→❝ ❣✐→ trà ①➜♣ ①➾ y1 , y2 , y3 , ✭❣✐→ trà ❝❤➼♥❤ ①→❝ ❧➔ y(x1 ), y(x2 ), y(x3 ), t↕✐ ❝→❝ ✤✐➸♠ x1 , x2 , x3 , ✮✳ ◆➳✉ ❝→❝ ✤✐➸♠ ❝❤✐❛ xn , n = 0, 1, 2, ✳ ❝➔♥❣ ♥❤✐➲✉ t❤➻ t❛ ❝➔♥❣ ❝â t➼♥❤ ❤➻♥❤ ↔♥❤ ❣➛♥ ✤ó♥❣ ❝õ❛ ❤➔♠ ♥❣❤✐➺♠ y(x)✳ ♠ët sè t trữớ ủ ữợ tự ❧➔ xn+1 −xn = h, n = 0, 1, 2, 3, ✳❚ø ❦❤❛✐ tr✐➸♥ ❚❛②❧♦r✱ ❣✐ú ❧↕✐ ❤❛✐ sè ❤↕♥❣ ✤➛✉ t❛ ❝â✿ y (ξ) 0) y(x1 ) = y(x0 ) + y (x 1! (x1 − x0 ) + 2! (x1 − x0 ) y(x1 ) = y0 + hf (x1!0 ,y0 ) + 0(h2 )✳ ❱➟② t❛ ❝â t❤➸ ❝â ✿ yn+1 ≈ yn + hf (xn ,yn ) ✳ 1! ❚❛ ♥❤➟♥ ✤÷đ❝ ❝ỉ♥❣ t❤ù❝ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r ♥❤÷ s❛✉ ✿ yn+1 = yn + hf (xn , yn ) ✭✷✳✸✮ P❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r ✭ð t ữ õ ỵ tữ sỷ ❞ö♥❣ ♠→② t➼♥❤ ✤➸ ✈➩ ♥➯♥ ❊✉❧❡r t➻♠ ♥❣❤✐➺♠ ❜➡♥❣ số ữợ tỹ tứ (xn , yn ) ✤➳♥ (xn+1 , yn+1 ) ❚r♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ dy dx = f (x, y) ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ y(x0 ) = y0 trữợ t t ữợ sỷ tứ ữợ tứ (x0 , y0 ) s ữợ s ❝â ✤✐➸♠ (xn , yn )✳ (xn , yn ) s❛♥❣ ✤✐➸♠ (xn+1 , yn+1 ) ✤÷đ❝ ①→❝ ✤à♥❤ ❦✐❛✳ ●✐↔ sû✱ ❜➢t ✤➛✉ ✈ỵ✐ ✤✐➸♠ ✤➛✉ ❱✐➺❝ t❤ü❝ ❤✐➺♥ ✈➩ tø ✤✐➸♠ q✉❛ ❝ỉ♥❣ t❤ù❝ ✭✷✳✸✮ ✈➔ ✤÷đ❝ ♠ỉ t↔ ð ❤➻♥❤ ✭✷✳✷✮✳ (xn , yn ) ❧➔ m = f (xn , yn )✳ ❱➻ ♠é✐ t❤❛② ✤ê✐ h tứ xn xn+1 tữỡ ự ợ sỹ t❤❛② ✤ê✐ ❝õ❛ mh = hf (xn , yn ) tø yn ✤➳♥ yn+1 ♥➯♥ ❝→❝ tå❛ ✤ë ❝õ❛ ✤✐➸♠ ợ (xn+1 , yn+1 ) ữủ ổ ố ữủ q tự ữợ tr♦♥❣ ❤➺ tå❛ ✤ë ✤➣ ❝❤♦ ✿ xn+1 = xn + h; yn+1 = yn + hf (xn , yn )✳ ✶✵ h = 0.1 ❝❤ó♥❣ t❛ ❝â ✿ y1 = y0 + 0.1x0 y0 = + 0.1(0 · 2) = y2 = y1 + 0.1x1 y1 = + 0.1(0.1 · 2) = 2.02 y3 = y1 + 0.1x2 y2 = 2.02 + 0.1(0.2 · 2.02) = 2.0604 ❱ỵ✐ ✳✳✳ ❱ỵ✐ ❤ ❂ ✵✳✵✶✱ ❝❤ó♥❣ t❛ ❝â ✿ y1 = y0 + 0.01x0 y0 = + 0.01(0 · 2) = y2 = y1 + 0.01x1 y1 = + 0.01(0.01 · 2) = 2.0002 y3 = y2 + 0.01x2 y2 = 2.0002 + 0.01(0.02 · 2.0002) = 2.0006 ✳✳✳ ❚ø ❝→❝ ❣✐→ trà t➻♠ ✤÷đ❝ t❛ ❧➟♣ ❜↔♥❣ t❤➸ ❤✐➺♥ ❝→❝ ❣✐→ trà ①➜♣ ①➾ ✈➔ ❝❤➼♥❤ ①→❝✳ ❚✉② ♥❤✐➯♥ ❜↔♥❣ ❣✐→ trà ❝â ❞ú ❧✐➺✉ ❦❤→ ❞➔✐ ♥➯♥ t❛ ❝â t❤➸ ❝❤✐❛ [0, 12 ] t❤➔♥❤ ✺ ✤å❛♥ ♥❤ä ✤➲✉ ✈➔ ❧➜② ❣✐→ trà t↕✐ ❝→❝ ♠è❝ ❝❤✐❛ ✤➸ ❧➟♣ ❜↔♥❣ ✭✷✳✺✮✳ ① ●❚❳❳ ❝õ❛ ② ù♥❣ ✈ỵ✐ ❤ ❂ ✵✳✶ ●❚❳❳ ❝õ❛ ② ù♥❣ ✈ỵ✐ ❤ ❂ ✵✳✵✶ ●✐→ trà t❤ü❝ ❝õ❛ ② ✵ ✷✳✵✵✵✵ ✷✳✵✵✵✵ ✷✳✵✵✵✵ ✵✳✶ ✷✳✵✵✵✵ ✷✳✵✵✽✹ ✷✳✵✶✵✵ ✵✳✷ ✷✳✵✷✵✵ ✷✳✵✸✽✵ ✷✳✵✹✵✹ ✵✳✸ ✷✳✵✻✵✹ ✷✳✵✽✸✽ ✷✳✵✾✷✵ ✵✳✹ ✷✳✶✷✷✷ ✷✳✶✺✻✽ ✷✳✶✻✻✻ ✵✳✺ ✷✳✷✵✼✶ ✷✳✷✺✹✼ ✷✳✷✻✻✸ ❇↔♥❣ ✷✳✺✿ ❳➜♣ ①➾ ❊✉❧❡r ❜➔✐ t♦→♥ y = xy ❀ y(0) = ✈ỵ✐ h = 0.1✱ h = 0.01 ✈➔ ❣✐→ trà ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ✣➸ ❣✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ♠➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❦❤ỉ♥❣ ❱➼ ❞ư ✷✳✼✿ ♣❤↔✐ ❧➔ ❝→❝ ❤➔♠ ❝ì ❜↔♥ t❛ sû ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r✳ ❉ị♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r ✤➸ t➻♠ ♥❣❤✐➺♠ ❜➔✐ t♦→♥ ❣✐→ trà ❜❛♥ ✤➛✉ ✿ dy dx = x2 − y , y(0) = h = 0.1, h = 0.2 tr♦♥❣ ✤♦↕♥ [0, 1] 2 ❚❛ ❝â f (x, y) = x − y ❀ ♥➯♥ →♣ ❞ư♥❣ ❱ỵ✐ ❝ỉ♥❣ t❤ù❝ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r ❧➔✿ yn+1 = yn + h((xn )2 − (yn )2 ) ✷✶ h = 0.1 t❛ ❝â y1 = y0 + 0.1((x0 )2 − (y0 )2 ) = 0.9 y2 = y1 + 0.1((x1 )2 − (y1 )2 ) = 0.82 y3 = y2 + 0.1((x2 )2 − (y2 )2 ) = 0.757 ❱ỵ✐ ✳✳✳ h = 0.2 t❛ ❝â y1 = y0 + 0.2((x0 )2 − (y0 )2 ) = 0.8 y2 = y1 + 0.2((x1 )2 − (y1 )2 ) = 0.68 y3 = y2 + 0.2((x2 )2 − (y2 )2 ) = 0.62 ợ ứ tr t ữủ t❛ ❧➟♣ ❜↔♥❣ t❤➸ ❤✐➺♥ ❝→❝ ❣✐→ trà ①➜♣ ①➾ ✈➔ ❝❤➼♥❤ ①→❝✳ ❚✉② ♥❤✐➯♥ ❜↔♥❣ ❣✐→ trà ❝â ❞ú ❧✐➺✉ ❦❤→ ❞➔✐ ♥➯♥ t❛ ❝â t❤➸ ❝❤✐❛ [0, 1] t❤➔♥❤ ✺ ✤å❛♥ ♥❤ä ✤➲✉ ✈➔ ❧➜② ❣✐→ trà t↕✐ ❝→❝ ♠è❝ ❝❤✐❛ ✤➸ ❧➟♣ ❜↔♥❣ ✭✷✳✻✮✳ ① ●❚❳❳ ❝õ❛ ② ù♥❣ ✈ỵ✐ ❤ ❂ ✵✳✶ ●❚❳❳ ❝õ❛ ② ù♥❣ ✈ỵ✐ ❤ ❂ ✵✳✷ ✵ ✶ ✶ ✵✳✷ ✵✳✽✷ ✵✳✽ ✵✳✹ ✵✳✼✵✽ ✵✳✻✽ ✵✳✻ ✵✳✻✺✹ ✵✳✻✶✾ ✵✳✽ ✵✳✻✺✹ ✵✳✻✶✺ ✶ ✵✳✼✵✼ ✵✳✻✻✼ dy ❇↔♥❣ ✷✳✻✿ ❳➜♣ ①➾ ❊✉❧❡r ❜➔✐ t♦→♥ dx = x2 − y , y(0) = 1✈ỵ✐ h = 0.1✱ h = 0.2 ❱➼ ❞ö ✷✳✽✿ ❳➨t ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ✿ x = −x + 8y; x(0) = 1, y = x + y; y(0) = ợ ữợ h = 0.05 t ①➾ ♥❣❤✐➺♠ ①✱ ② t↕✐ t = 0.04✳ ❙♦ s→♥❤ ✈ỵ✐ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝✳ ❙û ❞ư♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ t❛ t➻♠ ✤÷đ❝ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ❝õ❛ ❜➔✐ t♦→♥ ✤➣ ❝❤♦ ❧➔✿ −3t + 35 e3t , x(t) = −2 e y(t) = 16 e−3t + 65 e3t Ð ✤➙②✿ ✷✷ f (x, y) = −x + 8y; x0 = 1, g(x, y) = x + y; y0 = ⑩♣ ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ ✭✷✳✼✮ t❛ ❝â✿ xn+1 = xn + h(−xn + 8yn )✱ yn+1 = yn + h(xn + yn ) ợ ữợ t ❝â✿ x1 = x0 + 0.01(−x0 + 8y0 ) = 1.07, y1 = y0 + 0.01(x0 + y0 ) = 1.02 x2 = x1 + 0.01(−x1 + 8y1 ) = 1.1409, y2 = y1 + 0.01(x1 + y1 ) = 1.0195 x3 = x2 + 0.01(−x2 + 8y2 ) = 1.2111, y3 = y2 + 0.01(x2 + y2 ) = 1.0411 x4 = x3 + 0.01(−x3 + 8y3 ) = 1.3065, y4 = y3 + 0.01(x3 + y3 ) = 1.0636 ❈→❝ ❣✐→ trà ❝❤➼♥❤ ①→❝ t↕✐ t ❂ ✵✳✵✹ ❧➔✿ x(0.04) ≈ 1.2879 y(0.04) ≈ 1.0874 ✷✳✺ ×❯ ✣■➎▼✲ ❚➑◆❍ ✃◆ ✣➚◆❍ ❈Õ❆ P❍×❒◆● P❍⑩P ❳❻P ❳➓ ❊❯▲❊❘ ❚❘❖◆● P❍×❒◆● ❚❘➐◆❍ ❱■ P❍❹◆ ❚❍×❮◆● ❳➨t ❜➔✐ t♦→♥ ✿ ●✐↔ sû ❣✐→ trà ❜❛♥ ✤➛✉ ❝❤♦ |(y0 )d − (y0 )g | ≤ δ ✳ dy = f (x, y); y(x0 ) = y0 dx d ✤ó♥❣ ❧➔ (y0 ) ✈➔ ❣✐→ trà ❣➛♥ ✤ó♥❣ ❝õ❛ ♥â ❧➔ (y0 )g s❛♦ ❚❛ ①➨t s❛✐ sè ❜❛♥ ✤➛✉ ❝â ❜à s ữợ ổ õ (yi+1 )d = (yi )d + hf (xi , (yi )d )✱ (yi+1 )g = (yi )g + hf (xi , (yi )g )✱ =⇒ |(yi+1 )d − (yi+1 )g | ≤ |(yi )d − (yi )g | + h|f (xi , (yi )d ) − f (xi , (yi )g )| ≤ (1 + Lh)|(yi )d − (yi )g | ❱➟② |(yi+1 )d − (yi+1 )g | ≤ (1 + Lh)|(yi )d − (yi )g ✱ ✈ỵ✐ ♠å✐ ✐✳ ❚ø ✤â s✉② r❛✿ |(yn )d − (yn )g | ≤ (1 + Lh)|(yn−1 )d − (yn−1 )g | ≤ (1 + Lh)2 |(yn−2 )d − (yn−2 )g | ≤ ≤ (1 + Lh)n |(y0 )d − (y0 )g | ≤ (1 + Lh)n δ = enLh δ = eL(xn −x0 ) δ ◆❤÷ ✈➟② ♥➳✉ ❣✐→ trà ❜❛♥ ✤➛✉ ♠➢❝ s❛✐ sè t❤➻ s❛✐ sè ❦❤æ♥❣ t➠♥❣ ❧➯♥ s ữợ õ ữỡ r ê♥ ✤à♥❤✱ ✤➙② ❧➔ ÷✉ ✤✐➸♠ ♥ê✐ trë✐ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r✳ ✷✳✻ ❍❸◆ ❈❍➌ ❚❘❖◆● P❍×❒◆● P❍⑩P ❊❯▲❊❘ ❈❤ó♥❣ t❛ ✤➣ ①➨t t➼♥❤ ê♥ ✤à♥❤ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r tr♦♥❣ ♠ư❝ ✷✳✺✳✶✱ t❛ t❤➜② ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r ❧➔ ê♥ ✤à♥❤✳ ❚✉② ♥❤✐➯♥ ❝â ♠ët sè ❜➔✐ t♦→♥ ❣✐→ trà ✤➛✉ ❧↕✐ ❦❤ỉ♥❣ ❣✐↔✐ q✉②➳t tèt ♥❤÷ t❤➳✳ ❚❛ ①➨t ✈➼ ❞ö ✷✳✺ s❛✉ ✤➸ t➻♠ ❤✐➸✉ ❱➼ ❞ö ✷✳✾ ✿ t❤➯♠✳ ❉ị♥❣ ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ❊✉❧❡r ✤➸ t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛ ❜➔✐ t♦→♥ ❣✐→ trà ✤➛✉ ✿ dy = x2 + y ; y(0) = dx ✱ tr➯♥ ✤♦↕♥ ❬✵✱ ✶❪✳ ❱ỵ✐ ❜➔✐ t♦→♥ ♥➔② f (x, y) = x2 + y t❛ →♣ ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r ❧➔ ✿ yn+1 = yn + h((xn )2 + (yn )2 ) h = 0.1 ❝❤ó♥❣ t❛ ❝â ✿ y1 = + 0.1 · (02 + 12 ) = 1.1, y2 = 1.1 + 0.1 · (0.12 + 1.12 ) = 1.222, y3 = 1.222 + 0.1 · (0.22 + 1.2222 ) = 1.3753, ❱ỵ✐ ✳✳✳ ✷✹ h = 0.02✱ h = 0.005✱ h = 0.1✱ h = 0.02✱ h = 0.005✳ ❚÷ì♥❣ tỹ ổ tự r ữợ t õ t t q ợ ữợ ❳❡♠ ❜↔♥❣ ✭✷✳✼✮ t❛ t❤➜② sü ✑ê♥ ✤à♥❤✑ ❝õ❛ q✉→ tr trữợ ổ ② ù♥❣ ✈ỵ✐ ❤ ❂ ✵✳✶ ●❚❳❳ ❝õ❛ ② ù♥❣ ✈ỵ✐ ❤ ❂ ✵✳✵✷ ●❚❳❳ ❝õ❛ ② ù♥❣ ✈ỵ✐ ❤ ❂ ✵✳✵✵✺ ✵ ✶ ✶ ✶ ✵✳✶ ✶✳✶✵✵✵ ✶✳✶✵✽✽ ✶✳✶✶✵✽ ✵✳✷ ✶✳✷✷✷✵ ✶✳✷✹✺✽ ✶✳✷✺✶✷ ✵✳✸ ✶✳✸✼✺✸ ✶✳✹✷✹✸ ✶✳✹✸✺✼ ✵✳✹ ✶✳✺✼✸✺ ✶✳✻✻✺✽ ✶✳✻✽✽✷ ✵✳✺ ✶✳✽✸✼✶ ✷✳✵✵✼✹ ✷✳✵✺✶✷ ✵✳✻ ✷✳✶✾✾✺ ✷✳✺✷✵✶ ✷✳✻✶✵✹ ✵✳✼ ✷✳✼✶✾✸ ✸✳✸✻✶✷ ✸✳✺✼✵✻ ✵✳✽ ✸✳✺✵✼✽ ✹✳✾✻✵✶ ✺✳✺✼✻✸ ✵✳✾ ✹✳✽✵✷✸ ✾✳✵✵✵✵ ✶✷✳✷✵✻✶ ✶ ✼✳✶✽✾✺ ✸✵✾✶✻✼ ✶✺✵✷✳✷✵✾✵ dy ❇↔♥❣ ✷✳✼✿ ❳➜♣ ①➾ ❊✉❧❡r ❜➔✐ t♦→♥ dx = x2 + y ; y(0) = ❝á♥ ♥ú❛✳ ❘ã r➔♥❣ r➡♥❣ ❝â ❣➻ ✤â ❜➜t ê♥ ð ❣➛♥ x = 1✱ ✈➻ ✈➟② t❛ s➩ t➻♠ ♠✐➲♥ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✤÷đ❝ ✤÷❛ r❛✳ ❙û ❞ư♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ t❛ ✈➩ ✤÷đ❝ ♠✐➲♥ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ dy dx = x2 + y ũ ợ ữớ q✉❛ ✤✐➸♠ ✭✵✱ ✶✮✳ ❍➻♥❤ ✭✷✳✽✮ t❤➸ ❤✐➺♥ ♠✐➲♥ ❝❤ù❛ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❚ø ❤➻♥❤ ✈➩ t❛ t❤➜② ✤÷í♥❣ ❝♦♥❣ ❝â t✐➺♠ ❝➟♥ ✤ù♥❣ ð ❞ị ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r ❝❤♦ ♥❣❤✐➺♠ ð ❣➛♥ x = 1✱ dy dx = x2 + y ❣➛♥ x = 0.97✳ ▼➦❝ ♥❤÷♥❣ ổ tỗ t tr ❬✵✱✶❪✳ ❍ì♥ ♥ú❛✱ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r ❦❤ỉ♥❣ t❤➸ t❤❡♦ ❦à♣ ✤ë ❜✐➳♥ t❤✐➯♥ ♥❤❛♥❤ ❝õ❛ y(x) ❦❤✐ ① ❞➛♥ ✤➳♥ ❣✐→ trà t✐➺♠ ❝➟♥ ✤ù♥❣✳ ◗✉❛ ✈➼ ❞ö ✷✳✾ ❝❤ù♥❣ tä ✈➝♥ ❝â ♥❤ú♥❣ ❝↕♠ ❜➞② tr♦♥❣ ✈✐➺❝ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r✳ ❘ã r➔♥❣ ❦❤ỉ♥❣ t❤➸ t➻♠ ♥❣❤✐➺♠ ①➜♣ ①➾ tr➯♥ ♠ët ✤♦↕♥ ♠➔ tr➯♥ õ ổ tỗ t tr õ ❦❤ỉ♥❣ ❞✉② ♥❤➜t✮✳ ◆❣÷í✐ t❛ ❦❤ỉ♥❣ t❤➸ ❝❤➜♣ ♥❤➟♥ ❝→❝ t q ữỡ r ợ ởt số ữợ ố ữ tr t dy = x2 + y2; y(0) = ❍➻♥❤ ✷✳✽✿ dx q ữợ tự ợ ữợ ỡ ❤✴✺✱ ❤✴✶✵✮ s➩ ❝❤♦ ❝→❝ ❦➳t q✉↔ t❤➼❝❤ ❤ñ♣ tø ✤â ❞ü ✤♦→♥ ✤ë ❝❤➼♥❤ ①→❝✳ ✷✳✼ P❍⑩❚ ❚❘■➎◆ ❱⑨ ❙❖ ❙⑩◆❍ P❍×❒◆● P❍⑩P ❊❯▲❊❘ ❱❰■ ▼❐❚ ❙➮ P❍×❒◆● P❍⑩P ❑❍⑩❈ ❱ỵ✐ ♥❤ú♥❣ ❤↕♥ ❝❤➳ ✈➲ ♠➦t s❛✐ sè ❧ỵ♥✱ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r t✉② ❝â ÷✉ ✤✐➸♠ ❧➔ r➜t ✤ì♥ ❣✐↔♥✱ ❞➵ ❧➟♣ tr➻♥❤ ♥❤÷♥❣ ❝❤➾ ❞ị♥❣ ✤➸ t➻♠ ❧í✐ ❣✐↔✐ t❤ỉ ❝õ❛ ❜➔✐ t♦→♥ ❈❛✉❝❤②✳ ❚ø ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r t❛ ❝â t❤➸ ♣❤→t tr✐➸♥ t❤➔♥❤ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r ❝↔✐ t✐➳♥✳ P❤÷ì♥❣ ♣❤→♣ ♥➔② sû ❞ư♥❣ ✤ì♥ ❣✐↔♥ ✈➔ ❝â t❤➯♠ t❤✉➟♥ ❧ñ✐ ❦✐➸♠ tr❛ ❤➺ t❤è♥❣ ✈è♥ ❝â tr♦♥❣ q tr t ữủ t sỹ ữợ ữủ ❝❤♦ ②✳ ❳➨t ❜➔✐ t♦→♥ ❈❛✉❝❤② ✿ dy dx = f (x, y); y(x0 ) = y0 ●✐↔ sû s❛✉ ợ ữợ ú t t ✤÷đ❝ ❣✐→ trà ①➜♣ ①➾ xn = xn + nh✳ ●å✐ un+1 ✭❦❤æ♥❣ ♣❤↔✐ y(xn+1 ) ✮ ❧➔ ❣✐→ trà ♥❣❤✐➺♠ t↕✐ xn+1 = xn + (n + 1)h✱ ❦❤✐ ✤â un+1 = yn + hf (xn , yn ) = yn + hk1 ✳✳✳ ð x + xn ✤➣ ✤÷đ❝ t➼♥❤ t♦→♥✳ ❱➟② t↕✐ s❛♦ t❛ ❝õ❛ ❣✐→ trà ✤ó♥❣ y(xn ) yn ❝õ❛ ♥❣❤✐➺♠✱ t↕✐ ✷✻ ❦❤ỉ♥❣ ❧➜② ❣✐→ trà tr✉♥❣ ❜➻♥❤ ❝õ❛ ❤❛✐ ✤ë ❞è❝ ♥➔② ✤➸ õ ữủ ố ỡ ị tữ t ữỡ r t ợ ❞✉♥❣ ❝ì ❜↔♥ ♥❤÷ s❛✉✿ ❈❤♦ ❜➔✐ t♦→♥ ❣✐→ trà ✤➛✉✿ dy dx = f (x, y); y(x0 ) = y0 Pữỡ r t ợ ữợ ỗ ổ tự tr ỗ k1 = f (xn , yn ) ✭✷✳✽✮ un+1 = yn + hk1 ✭✷✳✾✮ k2 = f (xn+1 , un+1 ) yn+1 = yn + h (k1 + k2 ) ❈→❝ ❝æ♥❣ t❤ù❝ ♥➔② ❞ò♥❣ ✤➸ t➼♥❤ ❝→❝ ❣✐→ trà ①➜♣ ①➾ ❝❤➼♥❤ ①→❝ ②✭①✮ t↕✐ ❝→❝ ✤✐➸♠ ✭✷✳✶✵✮ ✭✷✳✶✶✮ y1 , y2 , y3 ✱✳✳✳ ❝õ❛ ♥❣❤✐➺♠ x1 , x2 , x3 ✱✳✳✳ ❈æ♥❣ t❤ù❝ ❝✉è✐ ð ✭✷✳✶✶✮ ❝â t❤➸ ✈✐➳t ữợ yn+1 = yn + 21 k ợ k = k1 + k2 ❱➟② ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r ❝↔✐ t✐➳♥ ợ ữợ ỗ sỷ t un+1 = yn + hf (xn , yn ) ✭✷✳✶✷✮ ❙❛✉ ✤â ❤✐➺✉ ❝❤➾♥❤✿ yn+1 = yn + h [f (xn , yn ) + f (xn+1 , un+1 )] t✐➳♣ tö❝ ♣❤➨♣ ❧➦♣ ✤➸ t➼♥❤ ❝→❝ ❣✐→ trà ❣➛♥ ✤ó♥❣ y1 , y2 , y3 ✱✳✳✳ tỵ✐ yx1 , yx2 , , yxn ❝õ❛ ♥❣❤✐➺♠ t❤ü❝ sü ❜➔✐ t tr ộ ữợ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r ❝↔✐ t✐➳♥ ❝➛♥ t➼♥❤ ❤❛✐ ❣✐→ trà ❤➔♠ ❢✭①✱②✮✱ tr♦♥❣ ❧ó❝ ✤â ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r t❤ỉ♥❣ t❤÷í♥❣ ❝❤➾ ❝➛♥ t➼♥❤ ♠ët ❣✐→ trà✱ ✈➟② t❤➻ ❧✐➺✉ ❝↔✐ t✐➳♥ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r ❝â ♣❤ù❝ t↕♣ ✈➔ ❦❤ỉ♥❣ ❤ú✉ ➼❝❤ ❦❤ỉ♥❣❄ ❱➼ ❞ư ✷✳✶✵✿ ❚❛ s➩ ✤✐ t➻♠ tr tổ q ữợ ❚➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛ ❜➔✐ t♦→♥ ❣✐→ trà ✤➛✉ ✿ dy dx = xy ; y(0) =1 ❚❤æ♥❣ q ữỡ r t ợ h = 0.1 s s ợ tr ữỡ ♣❤→♣ ❊✉❧❡r ð ❱➼ ❞ö ✭✷✳✷✮ ✈➔ ❣✐→ trà ♥❣❤✐➺♠ ữỡ r t ợ ❝ỉ♥❣ t❤ù❝ tr➯♥ ✈ỵ✐ x + y✱ t❛ ❝â✿ ✷✼ f (x, y) = yn+1 un+1 = yn + hf (xn , yn )✳ = yn + h 12 [f (xn , yn ) + f (xn+1 , yn+1 )]✳ h = 0.1 t❛ ✤÷đ❝✿ u1 = + 0.1( 0.1 ) =1 0.1.1 y1 = + 0.1 12 [ 0.1 + ] = 1.0025 u2 = 1.0025 + 0.1( 0.1.1.0025 ) = 1.0075 + 0.2.1.0075 ] = 1.01 y2 = 1.0025 + 0.1 12 [ 0.1.1.0025 2 ❱ỵ✐ ✳✳✳ ① P♣ ❊✉❧❡r ✈ỵ✐ ❤❂✵✳✶ P♣ ❊✉❧❡r ✈ỵ✐ ❤❂✵✳✵✺ P♣ ❊✉❧❡r ❝↔✐ t✐➳♥ ✈ỵ✐ ❤❂✵✳✶ ●✐→ trà t❤ü❝ t➳ ❝õ❛ ② ✵ ✶ ✶ ✶ ✶ ✵✳✶ ✶ ✶✳✵✵✶ ✶✳✵✵✸ ✶✳✵✵✸ ✵✳✷ ✶✳✵✵✺ ✶✳✵✵✼ ✶✳✵✶ ✶✳✵✶ ✵✳✸ ✶✳✵✶✺ ✶✳✵✶✾ ✶✳✵✷✸ ✶✳✵✷✹ ✵✳✹ ✶✳✵✸✵ ✶✳✵✸✺ ✶✳✵✹✵✼ ✶✳✵✹✽ ✵✳✺ ✶✳✵✺✶ ✶✳✵✺✽ ✶✳✵✻✹ ✶✳✵✻✺ ✵✳✻ ✶✳✵✼✼ ✶✳✵✽✺ ✶✳✵✾✹ ✶✳✵✾✹ ✵✳✼ ✶✳✶✵✾ ✶✳✶✷✵ ✶✳✶✸✵ ✶✳✶✸✵ ✵✳✽ ✶✳✶✹✽ ✶✳✶✻✶ ✶✳✶✼✸ ✶✳✶✼✸ ✵✳✾ ✶✳✶✾✹ ✶✳✷✵✾ ✶✳✷✷✹ ✶✳✷✷✹ ✶ ✶✳✷✹✽ ✶✳✷✻✻ ✶✳✷✽✹ ✶✳✷✽✹ dy ❇↔♥❣ ✷✳✽✿ ❳➜♣ ①➾ ❊✉❧❡r ❜➔✐ t♦→♥ dx = xy ; y(0) = ❚r♦♥❣ ❜↔♥❣ ✭✷✳✽✮ s♦ s→♥❤ ❦➳t q✉↔ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r t❤ỉ♥❣ t❤÷í♥❣ ữỡ r t ũ ữợ h = 0.1 t❤➻ s❛✐ sè tr♦♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r t❤ỉ♥❣ t❤÷í♥❣ ✤➸ t➼♥❤ ②✭✶✮ ❧➔ ✵✳✵✼✷✺ ❝á♥ s❛✐ sè ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r ❝↔✐ t✐➳♥ ❧➔ ✵✳✵✵✷✹✳ ❚❤ü❝ t➳✱ ✈➼ ❞ö t ữỡ r t ợ ỡ s ợ ữỡ r tổ tữớ ợ ữỡ r t ♣❤➨♣ t➼♥❤ tr♦♥❣ ❦❤✐ ❝➛♥ ✤➳♥ ✷✵✵ ♣❤➨♣ t➼♥❤ ❧♦↕✐ õ ợ ữỡ r tổ tữớ ữ ữỡ ♣❤→♣ ❊✉❧❡r ❝↔✐ t✐➳♥ ✈ø❛ ❝❤➼♥❤ ①→❝ ❤ì♥ ♥❤✐➲✉ ❧↕✐ ❝❤➾ ❝➛♥ 10 ❦❤è✐ ❧÷đ♥❣ t➼♥❤ t♦→♥✳ ◆â✐ ❝❤✉♥❣✱ ợ ữỡ r t t õ t ①➙② ❞ü♥❣ ♥➯♥ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r ❝↔✐ t✐➳♥✲♠ët ❝ỉ♥❣ ❝ư ✤➸ ❣✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝❤➼♥❤ ①→❝ ❤ì♥✳ ữỡ ệ P Pì P❍⑩P ❳❻P ❳➓ ❊❯▲❊❘ ✸✳✶ ●■❰■ ❚❍■➏❯ ❱➋ P❍❺◆ ▼➋▼ r tr ữỡ ứ rỗ ❝❤ó♥❣ t❛ ❝â ✤➲ ❝➟♣ ✤➳♥ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛✱ ✈➟② ▼❛t❤❡♠❛t✐❝❛ ❧➔ ❣➻ ✈➔ ♥â ❝â ❝ỉ♥❣ ❞ư♥❣ r❛ s❛♦ tr♦♥❣ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ♥➔②✳ ▼❛t❤❡♠❛t✐❝❛ ❧➛♥ ✤➛✉ t✐➯♥ ✤÷đ❝ ❤➣♥❣ ❲♦❧❢r❛♠ ❘❡s❡❛r❝❤ ♣❤→t ❤➔♥❤ ✈➔♦ ♥➠♠ ✶✾✽✽ ❧➔ ♠ët ❤➺ t❤è♥❣ ♥❤➡♠ t❤ü❝ ❤✐➺♥ ❝→❝ t➼♥❤ t♦→♥ t♦→♥ ❤å❝ tr➯♥ ♠→② t➼♥❤ ✤✐➺♥ tû✳ ◆â ❧➔ ♠ët tê ủ t t ỵ t t số ỗ t ổ ỳ tr t✐♥❤ ✈✐✳ ▲➛♥ ✤➛✉ t✐➯♥ ❦❤✐ ✈❡rs✐♦♥ ✶ ❝õ❛ ▼❛t❤❡♠❛t✐❝❛ ✤÷đ❝ ♣❤→t ❤➔♥❤✱ ♠ư❝ ✤➼❝❤ ❝❤➼♥❤ ❝õ❛ ♣❤➛♥ ♠➲♠ ♥➔② ❧➔ ✤÷❛ ✈➔♦ sû ❞ư♥❣ ❝❤♦ ❝→❝ ♥❣➔♥❤ ❦❤♦❛ ❤å❝ t ỵ ổ t ũ ợ tớ ❣✐❛♥ ▼❛t❤❡♠❛t✐❝❛ trð t❤➔♥❤ ♣❤➛♥ ♠➲♠ q✉❛♥ trå♥❣ tr♦♥❣ ♥❤✐➲✉ ❧➽♥❤ ✈ü❝ ❦❤♦❛ ❤å❝ ❦❤→❝✳ ◆❣➔② ♥❛②✱ ▼❛t❤❡♠❛t✐❝❛ ❦❤æ♥❣ ♥❤ú♥❣ ✤÷đ❝ sû ❞ư♥❣ tr♦♥❣ ❝→❝ ♥❣➔♥❤ ❦❤♦❛ ❤å❝ tü ♥❤✐➯♥ ữ t ỵ s t õ ổ ♥❣❤➺ ♠➔ ♥â ❝á♥ trð t❤➔♥❤ ♠ët ♣❤➛♥ ♠➲♠ q✉❛♥ trå♥❣ tr♦♥❣ ❝→❝ ♥❣➔♥❤ ❦❤♦❛ ❤å❝ ①➣ ❤ë✐ ❝ơ♥❣ ♥❤÷ ❦✐♥❤ t➳✳ ❚r♦♥❣ ❝ỉ♥❣ ♥❣❤➺ ♥❣➔② ♥❛② ♥❣÷í✐ t❛ sû ❞ư♥❣ ▼❛t❤❡♠❛t✐❝❛ tr♦♥❣ ❝ỉ♥❣ t→❝ t❤✐➳t ❦➳ ✈➔ r➜t ♥❤✐➲✉ ù♥❣ ❞ư♥❣ ❦❤→❝ ❝õ❛ ♣❤➛♥ ♠➲♠ ♥➔②✳ ❙è ♥❣÷í✐ sû ❞ö♥❣ ▼❛t❤❡♠❛t✐❝❛ ♥❣➔② ❝➔♥❣ t➠♥❣✳ ❚❤❡♦ sè ❧✐➺✉ ✷✾ ❣➛♥ ✤➙②✱ t➜t ❝↔ ❝→❝ ❝æ♥❣ t② ❝â tr♦♥❣ ❋♦rt✉♥❡ ✺✵✱ ❤➛✉ ❤➳t ✶✺ ❜ë ❝❤õ ❝❤èt ❝õ❛ ❝❤➼♥❤ ♣❤õ ❍♦❛ ý trữớ ợ t tr t ❣✐ỵ✐ ✤➲✉ sû ❞ư♥❣ ▼❛t❤❡♠❛t✐❝❛✳ ❚→❝ ❣✐↔ ❝õ❛ ▼❛t❤❡♠❛t✐❝❛ ❧➔ ❙t❡♣❤❡♥ ❲♦❧❢r❛♠✱ ♥❣÷í✐ ✤÷đ❝ ①❡♠ ❧➔ ♥❤➔ s→♥❣ t↕♦ q✉❛♥ trå♥❣ ♥❤➜t tr♦♥❣ ❧➽♥❤ ✈ü❝ t➼♥❤ t♦→♥ ❦❤♦❛ ❤å❝ ✈➔ ❦ÿ t❤✉➟t ♥❣➔② ♥❛②✳ ➷♥❣ s✐♥❤ ♥➠♠ ✶✾✺✾ t↕✐ ▲♦♥❞♦♥ ✈➔ ❤å❝ t↕✐ ❝→❝ tr÷í♥❣ ❊t♦♥✱ ❖①❢♦r❞ ✈➔ ❈❛❧t❡❝❤✳ ➷♥❣ ❜➢t ✤➛✉ ♣❤→t tr✐➸♥ ▼❛t❤❡♠❛t✐❝❛ ✈➔♦ ♥➠♠ ✶✾✽✻✳ ❱❡rs✐♦♥ ✤➛✉ t✐➯♥ ❝õ❛ ▼❛t❤❡♠❛t✐❝❛ ✤÷đ❝ ❝ỉ♥❣ ❜è ♥❣➔② ✷✸ t❤→♥❣ ✻ ♥➠♠ ✶✾✽✽✳ ❈ỉ♥❣ tr➻♥❤ ♥➔② ✤÷đ❝ ①❡♠ ❧➔ t❤➔♥❤ tü✉ ❝❤➼♥❤ tr♦♥❣ ❧➽♥❤ ✈ü❝ ❦❤♦❛ ❤å❝ t➼♥❤ t♦→♥ ❝ơ♥❣ ♥❤÷ ❧➔ ♠ët ♣❤➛♥ ♠➲♠ t♦→♥ ❤å❝ ♥ê✐ t✐➳♥❣ ✈➔♦ ❜➟❝ ♥❤➜t ❤✐➺♥ ♥❛②✳ ✸✳✷ Ù◆● ❉Ö◆● P❍❺◆ ▼➋▼ ▼❆❚❍❊▼❆❚■❈❆ ❈❍❖ P❍×❒◆● P❍⑩P ❳❻P ❳➓ ❊❯▲❊❘ ❱✐➺❝ ❧➟♣ tr➻♥❤ ✈✐ t➼♥❤ ✤➸ t❤ü❝ ❤✐➺♥ ♠ët t❤✉➟t sè s➩ ❧➔♠ ❝❤♦ ❦➼❝❤ tữợ tt t ữớ t s s↔♦ ❤ì♥✳ ❚✉② ♥❤✐➯♥ ✤➸ ❣✐↔✐ ♠ët ❜➔✐ t♦→♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r t❤ỉ♥❣ q✉❛ ❧➟♣ tr➻♥❤ ♠→② t➼♥❤ ✈➝♥ ❝❤÷❛ ✤÷đ❝ ❜✐➳t ✤➳♥ ♥❤✐➲✉ tr♦♥❣ s✐♥❤ ✈✐➯♥ ❝→❝ ❦❤è✐ ✤↕✐ ❤å❝✳ ❱➻ t❤➳ tr♦♥❣ ✤➲ t➔✐ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ♥❣❤✐➯♥ ❝ù✉ ù♥❣ ❞ư♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ✤➸ t➻♠ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ✈➔ ♥❣❤✐➺♠ ①➜♣ ữỡ tr tữớ ỗ t t❤ỉ♥❣ q✉❛ ❝→❝ ❣â✐ ❝➙✉ ❧➺♥❤ ✤➣ ✤÷đ❝ ❧➟♣ tr➻♥❤✳ ✣➸ t➻♠ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ❝õ❛ ❜➔✐ t♦→♥ dx dy = x − y; y(0) = −2 ❱ỵ✐ ❝→❝❤ ❣✐↔✐ ❜➡♥❣ t❛② t❤õ ❝ỉ♥❣ t❤ỉ♥❣ t❤÷í♥❣ t❛ ❧➔♠ ♥❤÷ s❛✉✿ y =x−y ⇒y +y =x ❚❛ ❝â✿ P (x) = 1, ⇒ P (x)dx = x q(x) = x I = q(x).e P (x)dx dx = = x.ex − ex dx = = x.ex − ex x.ex dx = ◆❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❧➔✿ ✸✵ y(x) = e− P (x)dx dx.( q(x).e P (x)dx dx + C) = = e−x (x.ex − ex + C) = = x − + C.e−x ▼➔ y(0) = −2✳ ❚❤❛② x = 0, y = −2 ✈➔♦ ♥❣❤✐➺♠ tê♥❣ q✉→t ✈ø❛ ⇒ −1 + C.e0 = −2 C = −1 t➻♠ t❛ ❝â✿ ❱➟② ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ❝õ❛ ❜➔✐ t♦→♥ ✤➣ ❝❤♦ ❧➔✿ y(x) = e−x (xex − ex 1) ữ ợ tt t ♠ð ♠ët ❢✐❧❡ ♠ỵ✐ ✈➔ ❣ã ❝➙✉ ❧➺♥❤✿ ❈❧❡❛r❬②✱ ①❪❀ ❉❙♦❧✈❡ ❬②✬❬①❪❂❂①✲②❬①❪✱ ②❬✵❪❂❂✲✷✱ ②❬①❪✱ ①❪ ✈➔ ♥❤➜♥ tê ❤ñ♣ ♣❤➼♠ s❤✐❢t ✈➔ ❡♥t❡r t❛ s➩ ✤÷đ❝ ❦➳t q✉↔ ♥❤÷ tr♦♥❣ ❤➻♥❤ ✭✸✳✶✮ ❍➻♥❤ ✸✳✶✿ ❚➻♠ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ❜➔✐ t♦→♥ ợ tt ỗ tớ t ụ ❝â t❤➸ t➻♠ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ tr♦♥❣ ❱➼ ❞ö ✷✳✽ ❤➻♥❤ ✭✸✳✷✮ ✸✶ ❍➻♥❤ ✸✳✷✿ ❚➻♠ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ❜➔✐ t♦→♥ ✈➼ ❞ö ✷✳✻ ❑❤✐ t❤ü❝ ❤✐➺♥ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ❊✉❧❡r t❛ ❝â t❤➸ ❧➟♣ tr➻♥❤ ởt tt t tỹ ữợ ởt ♥❤❛♥❤ ❝❤â♥❣ tr♦♥❣ ♥❤÷ tr♦♥❣ ❤➻♥❤ ✭✸✳✸✮ ✈➔ ❤➻♥❤ ✭✸✳✹✮ ❍➻♥❤ ✸✳✸✿ ▲➟♣ tr➻♥❤ ▼❛t❤❡♠❛t✐❝❛ t➻♠ ♥❣❤✐➺♠ ①➜♣ ①➾ ❜➔✐ t♦→♥ ð ✈➼ ❞ư ✷✳✶ ✈ỵ✐ ❤ ❂ ✵✳✺ ✸✷ ❍➻♥❤ ✸✳✹✿ ▲➟♣ tr➻♥❤ ▼❛t❤❡♠❛t✐❝❛ t➻♠ ♥❣❤✐➺♠ ①➜♣ ①➾ ❜➔✐ t♦→♥ ð ✈➼ ❞ư ✷✳✶ ✈ỵ✐ ❤ ❂ ✵✳✷ ❇➯♥ ❝↕♥❤ ✤â ✈✐➺❝ ❣✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤❡♦ ♣❤÷ì♥❣ ♣❤→♣ ❊✉✲ ❧❡r t❤ỉ♥❣ q✉❛ ♣❤➛♥ ♠➲♠ ♥➔② ❝ơ♥❣ ❦❤æ♥❣ q✉→ ♣❤ù❝ t↕♣✱ t❛ ❝â t❤➸ ①❡♠ ❤➻♥❤✭✸✳✺✮✳ ❍➻♥❤ ✸✳✺✿ ▲➟♣ tr➻♥❤ ▼❛t❤❡♠❛t✐❝❛ t➻♠ ♥❣❤✐➺♠ ①➜♣ ①➾ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ tr♦♥❣ ❱➼ ❞ư ✷✳✼ t❤❡♦ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r ✸✸ ❑➌❚ ▲❯❾◆ ✣➲ t➔✐ ✧P❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ❊✉❧❡r tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣✧ ✤➣ ✤↕t ✤÷đ❝ ♥❤ú♥❣ ❦➳t q✉↔ s❛✉✿ ❛✳ ✣➲ t➔✐ ✤÷❛ r❛ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r ❝ơ♥❣ ♥❤÷ ù♥❣ ❞ư♥❣ tø ✤â ✤÷❛ r❛ ♥❤ú♥❣ ❤↕♥ ❝❤➳ ✤➸ ❦❤➢❝ ♣❤ư❝ ✈➔ ♣❤→t tr✐➸♥ ♣❤÷ì♥❣ ♣❤→♣ ♥➔②✳ ❜✳ ✣➲ t➔✐ ✤➣ ✤÷❛ r❛ ♠ët sè ❝→❝ ✈➼ ❞ư ❝ơ♥❣ ♥❤÷ ❜➔✐ t➟♣ ✤➸ ♥➯✉ ❜➟t ❧➯♥ ÷✉ ❦❤✉②➳t ✤✐➸♠ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❊✉❧❡r✳ ❝✳ ✣➲ t➔✐ ✤➣ ✤÷❛ r❛ ♠ët ♣❤➛♥ ♠➲♠ ù♥❣ ❞ö♥❣ ♠❛♥❣ t➯♥ ▼❛t❤❡♠❛t✐❝❛ ❝â þ ♥❣❤➽❛ ❤ê trñ ✈✐➺❝ ❣✐↔✐ t♦→♥ ✈✐ ♣❤➙♥ ♠ët õ t õ ỵ ♥❣❤➽❛ t❤ü❝ t✐➵♥ ❧➔ ❝â t❤➸ ❧➔♠ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤♦ s✐♥❤ ✈✐➯♥ ❝→❝ ♥❣➔♥❤ ❚♦→♥✱ s✐♥❤ ✈✐➯♥ s÷ ♣❤↕♠✱ ❝û ♥❤➙♥ ❚♦→♥ Ù♥❣ ❉ö♥❣ tr♦♥❣ q✉→ tr➻♥❤ ❞↕② ✈➔ ❤å❝ ♠ỉ♥ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✈➔ ❣✐↔✐ t➼❝❤ sè✳ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❧➔ ♠ët ❜ë ♠ỉ♥ ❤å❝ rt ú ợ ữỡ t➼♥❤❀ ❞♦ ✈➟② ✤➲ t➔✐ ❝â ❦❤↔ ♥➠♥❣ ♠ð rë♥❣ ♥❣❤✐➯♥ ❝ù✉ t❤➯♠ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❦❤→❝✱ ✤➦❝ ❜✐➺t ❧➔ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ t➼♥❤ ①➜♣ ①➾✿ P❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ởt ữợ ữ ữỡ r t ữỡ tt ữỡ ữợ ữ ữỡ ♣❤→♣ ❆❞❛♠s✳ ❉♦ ❦❤↔ ♥➠♥❣ ❝õ❛ ❜↔♥ t❤➙♥ ❝á♥ ❤↕♥ ❝❤➳✱ ♠➦❝ ❞ò ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣ s♦♥❣ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ s❛✐ sât✱ t→❝ ❣✐↔ r➜t ữủ ỳ ỵ õ qõ qỵ ❝õ❛ ❝→❝ t❤➛② ❝æ ✈➔ ❝→❝ ❜↕♥ ✤➸ ✤➲ t➔✐ ♣❤→t tr✐➸♥ ❤ì♥✳ ✸✹ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❚ỉ♥ ❚➼❝❤ ⑩✐✭✷✵✵✶✮✳ P❤÷ì♥❣ ♣❤→♣ sè ✱ ◆①❜ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✳ ❬✷❪ P❤↕♠ ❑ý ❆♥❤✭✷✵✵✽✮✳ ●✐↔✐ t➼❝❤ sè✱ ◆①❜ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✳ ❬✸❪ ❚r➛♥ ❆♥❤ ❇↔♦✱ ◆❣✉②➵♥ ❱➠♥ ❑❤↔✐✱ P❤↕♠ ❱➠♥ ❑✐➲✉✱ ◆❣ỉ ❳✉➙♥ ❙ì♥ ✭✷✵✵✼✮✳ ●✐↔✐ t➼❝❤ sè✱ ◆❤➔ ①✉➜t ❜↔♥ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠✳ ❬✹❪ P❤❛♥ ✣➠♥❣ ❈➛✉✱ P❤❛♥ ❚❤à ❍➔ ✭✷✵✵✷✮✳ ●✐→♦ tr➻♥❤ ♣❤÷ì♥❣ ♣❤→♣ sè✱ ❍å❝ ✈✐➺♥ ❝ỉ♥❣ ♥❣❤➺ ❇÷✉ ❝❤➼♥❤ ❱✐➵♥ ❚❤æ♥❣✳ ❬✺❪ ◆❣✉②➵♥ ❍ú✉ ✣✐➸♥✱ ◆❣✉②➵♥ ▼✐♥❤ ❚✉➜♥ ✭✷✵✵✶✮✳ ▲❛t❡①✱ ◆①❜ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✳ ❬✻❪ ❊❞✈❛r❞s ❈✳❍❡♥❞r②✱ ❉❛✈✐❞ ❊✳P❡♥♥❡② ✭✷✵✵✼✮✳ ❚r❛ ❝ù✉ ✈➔ s♦↕♥ t❤↔♦ ❊❧❡♠❡♥t❛r② ❞✐❢❡r❡♥t✐❛❧ ❡q✉❛❞t✐♦♥s ✇✐t❤ ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s✱ Pr❡♥t✐❝❡ ❍❛❧❧✳ ❬✼❪ ◆✳❇❛❝✈❛❧♦♣✭✶✾✼✻✮✳ ❬✽❪ ❘♦❣❡r ▼❡t❤♦❞❡s ◆✉♠❜❡r✐q✉❡✳ ❈♦♦❦❡✭✶✾✽✹✮✳ ❚❤❡ ▼❛t❤❡♠❛t✐❝s ♦❢ ❙♦♥②❛ ❑♦✈❛❧❡✈s❦❛②❛✱ ❙♣r✐♥❣❡r✲ ❱❡r❧❛❣✳ ✸✺