Bài giảng toán cao cấp phần 1

Möc löc C�C KÞ HI�U 7 1 H�M SÈ NHI�U BI�N SÈ 9 1 1 Kh¡i ni»m mð �¦u 9 1 1 1 Khæng gian metric 9 1 1 2 �ành ngh¾a h m sè n bi¸n sè 10 1 1 3 Giîi h¤n cõa h m nhi u bi¸n 10 1 1 4 Sü li¶n töc cõa h m nhi. taài liệu cao đẳng đại học, tài liệu luận văn, giáo trình thạc sy, tiến sỹ, tài liệu THCS Bài giảng toán cao cấp phần 1

▼ư❝ ❧ư❝ ❈⑩❈ ❑Þ ❍■➏❯ ✼ ✶✳ ❍⑨▼ ❙➮ ◆❍■➋❯ ❇■➌◆ ❙➮ ✾ ✶✳✶✳ ✶✳✷✳ ✶✳✸✳ ❑❤→✐ ♥✐➺♠ ♠ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❑❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✶✳✷✳ ✣à♥❤ ♥❣❤➽❛ ❤➔♠ sè ♥ ❜✐➳♥ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✶✳✸✳ ●✐ỵ✐ ❤↕♥ ❝õ❛ ❤➔♠ ♥❤✐➲✉ ❜✐➳♥ ✶✳✶✳✹✳ ❙ü ❧✐➯♥ tö❝ ❝õ❛ ❤➔♠ ♥❤✐➲✉ ❜✐➳♥ ✣↕♦ ❤➔♠ r✐➯♥❣ ✈➔ ✈✐ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✷✳✶✳ ✣à♥❤ ♥❣❤➽❛ ✤↕♦ ❤➔♠ r✐➯♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✷✳✷✳ ❱✐ ♣❤➙♥ t♦➔♥ ♣❤➛♥ ✶✳✷✳✸✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✣↕♦ ❤➔♠ r✐➯♥❣ ✈➔ ✈✐ ♣❤➙♥ ❝➜♣ ❝❛♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✷✳✹✳ ❈ỉ♥❣ t❤ù❝ ❚❛②❧♦r ✤è✐ ✈ỵ✐ ❤➔♠ ♥❤✐➲✉ ❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ❈ü❝ trà ❝õ❛ ❤➔♠ ♥❤✐➲✉ ❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✸✳✶✳ ❈ü❝ trà tü ❞♦ ❝õ❛ ❤➔♠ ♥❤✐➲✉ ❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✸✳✷✳ ❈ü❝ trà ❝â ✤✐➲✉ ❦✐➺♥ ❝õ❛ ❤➔♠ ♥❤✐➲✉ ❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✸✳✸✳ ●✐→ trà ❧ỵ♥ ♥❤➜t ✈➔ ♥❤ä ♥❤➜t ❝õ❛ ❤➔♠ ♥❤✐➲✉ ❜✐➳♥ tr➯♥ ♠✐➲♥ ✤â♥❣✱ ❜à ❝❤➦♥ ✷✷ ❇➔✐ t➟♣ ❝❤÷ì♥❣ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳ ❚➑❈❍ P❍❹◆ ❑➆P ❱⑨ ❚➑❈❍ P❍❹◆ ✣×❮◆● ▲❖❸■ ■■ ✷✳✶✳ ✷✳✷✳ ✷✳✸✳ ✷✺ ✷✾ ❚➼❝❤ ♣❤➙♥ ❦➨♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✶✳✶✳ ✣à♥❤ ♥❣❤➽❛ ✷✾ ✷✳✶✳✷✳ ❈→❝❤ t➼♥❤ t➼❝❤ ♣❤➙♥ ❦➨♣ tr♦♥❣ ❤➺ tå❛ ✤ë ✣➲❝→❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✶✳✸✳ ✣ê✐ ❜✐➳♥ sè tr♦♥❣ t➼❝❤ ♣❤➙♥ ❦➨♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ Ù♥❣ ❞ö♥❣ ❝õ❛ t➼❝❤ ♣❤➙♥ ❦➨♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✳✷✳✶✳ Ù♥❣ ❞ư♥❣ ❤➻♥❤ ❤å❝ ✈➔ ❝ì ❤å❝ ❝õ❛ t➼❝❤ ♣❤➙♥ ❦➨♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚➼❝❤ ♣❤➙♥ ✤÷í♥❣ ❧♦↕✐ ❤❛✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✷✳✸✳✶✳ ✣à♥❤ ♥❣❤➽❛ ✈➔ t➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✷✳✸✳✷✳ ❈→❝❤ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✷✳✸✳✸✳ ❈æ♥❣ t❤ù❝ ●r❡❡♥ ✷✳✸✳✹✳ ✣✐➲✉ ❦✐➺♥ ✤➸ t➼❝❤ ♣❤➙♥ ✤÷í♥❣ ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✤÷í♥❣ ❧➜② t➼❝❤ ♣❤➙♥ ✳ ✳ ✳ ✺✹ ✷✳✸✳✺✳ ❚r÷í♥❣ ❤đ♣ ✤÷í♥❣ ❧➜② t➼❝❤ ♣❤➙♥ ❧➔ ♠ët ✤÷í♥❣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✳ ✳ ✳ ✳ ✳ ✺✻ ❇➔✐ t➟♣ ❝❤÷ì♥❣ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳ P❍×❒◆● ❚❘➐◆❍ ❱■ P❍❹◆ ✸✳✶✳ ✾ ✶✳✶✳✶✳ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ✶ ✹✾ ✺✽ ✻✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✸✳✶✳✶✳ ✣↕✐ ❝÷ì♥❣ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✸✳✶✳✷✳ P❤÷ì♥❣ tr➻♥❤ ❦❤✉②➳t ✻✹ ✸✳✶✳✸✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ♠ët ❝â ❜✐➳♥ sè ♣❤➙♥ ❧②✭P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ♠ët t→❝❤ ❜✐➳♥ ✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✷ ▼Ư❈ ▲Ư❈ ✸✳✶✳✹✳ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✤➥♥❣ ❝➜♣ ❝➜♣ ✶ ✭P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤✉➛♥ ♥❤➜t ❝➜♣ ✶✮ ✸✳✷✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ✸✳✶✳✺✳ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✸✳✶✳✻✳ P❤÷ì♥❣ tr➻♥❤ ❇❡❝♥✉❧❧② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✸✳✶✳✼✳ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ✶ t♦➔♥ ♣❤➛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✸✳✷✳✶✳ ✣↕✐ ❝÷ì♥❣ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✸✳✷✳✷✳ P❤÷ì♥❣ tr➻♥❤ ❦❤✉②➳t ✼✹ ✸✳✷✳✸✳ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ❤❛✐ ❝â ❤➺ sè t❤❛② ✤ê✐ ✸✳✷✳✹✳ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ❤❛✐ ❝â ❤➺ sè ❦❤æ♥❣ ✤ê✐ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ✷ ❇➔✐ t➟♣ ❝❤÷ì♥❣ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹ ✹✳ ▼❆ ❚❘❾◆ ✲ ✣➚◆❍ ❚❍Ù❈ ✲ ❍➏ P❍×❒◆● ❚❘➐◆❍ ❚❯❨➌◆ ❚➑◆❍ ✹✳✶✳ ✹✳✷✳ ✹✳✸✳ ✹✳✹✳ ✹✳✺✳ ▼❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✼ ✽✼ ✹✳✶✳✶✳ ❑❤→✐ ♥✐➺♠ ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✼ ✹✳✶✳✷✳ ▼ët sè ❞↕♥❣ ✤➦❝ ❜✐➺t ❝õ❛ ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✼ ✹✳✶✳✸✳ P❤➨♣ t♦→♥ tr➯♥ ♠❛ tr➟♥ ✽✾ ✹✳✶✳✹✳ ❇✐➳♥ ✤ê✐ ❝➜♣ tr➯♥ ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✣à♥❤ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✷✳✶✳ ✣à♥❤ ♥❣❤➽❛ ✹✳✷✳✷✳ ❚➼♥❤ ❝❤➜t ✹✳✷✳✸✳ ❚➼♥❤ ✤à♥❤ t❤ù❝ ❜➡♥❣ ❜✐➳♥ ✤ê✐ ❝➜♣ ✾✶ ✾✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺ ▼❛ tr➟♥ ♥❣❤à❝❤ ✤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✻ ✹✳✸✳✶✳ ✣à♥❤ ♥❣❤➽❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✻ ✹✳✸✳✷✳ ❚➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✻ ✹✳✸✳✸✳ ❚➻♠ ♠❛ tr➟♥ ♥❣❤à❝❤ ✤↔♦ ❜➡♥❣ ♣❤ö ✤↕✐ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✽ ✹✳✸✳✹✳ ❚➻♠ ♠❛ tr➟♥ ♥❣❤à❝❤ ✤↔♦ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ●❛✉ss✲❏♦r❞❛♥ ✾✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❍↕♥❣ ❝õ❛ ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✵ ✹✳✹✳✶✳ ✣à♥❤ ♥❣❤➽❛ ✹✳✹✳✷✳ ❚➻♠ ❤↕♥❣ ❝õ❛ ♠❛ tr➟♥ ❜➡♥❣ ❜✐➳♥ ✤ê✐ ❝➜♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✵ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✶ ✹✳✺✳✶✳ ✣à♥❤ ♥❣❤➽❛ ✹✳✺✳✷✳ ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❜➡♥❣ ♠❛ tr➟♥ ♥❣❤à❝❤ ✤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✷ ✹✳✺✳✸✳ ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❈r❛♠❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✷ ✹✳✺✳✹✳ ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ●❛✉ss ✹✳✺✳✺✳ ●✐↔✐ ✈➔ ❜✐➺♥ ❧✉➟♥ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❞ü❛ ✈➔♦ ✤à♥❤ ỵ rr ữỡ tr t t t t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✼ ❇➔✐ t➟♣ ❝❤÷ì♥❣ ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✹ ✳ ✳ ✶✵✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✽ ❆✳ P❍➆P ❚➑◆❍ ❱■✱ ❚➑❈❍ P❍❹◆ ❍⑨▼ ❙➮ ▼❐❚ ❇■➌◆ ❙➮ P❍➆P ❚➑◆❍ ❱■✱ ❚➑❈❍ P❍❹◆ ❍⑨▼ ❙➮ ▼❐❚ ❇■➌◆ ❙➮ ✶✷✶ ✶✷✶ ❆✳✶✳ ⑩♥❤ ①↕ ✈➔ ❤➔♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✶ ❆✳✶✳✶✳ ❈→❝ ✤à♥❤ ♥❣❤➽❛ ✈➲ →♥❤ ①↕ ✈➔ ❤➔♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✶ ❆✳✶✳✷✳ ❍➔♠ sè ❝➜♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✸ ❆✳✷✳ P❤➨♣ t➼♥❤ ✈✐ ♣❤➙♥ ❤➔♠ ♠ët ❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✼ ❆✳✷✳✶✳ ✣↕♦ ❤➔♠ ✈➔ ✈✐ ♣❤➙♥ ❝➜♣ ♠ët ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✼ ❆✳✷✳✷✳ ✣↕♦ ❤➔♠ ✈➔ ✈✐ ♣❤➙♥ ❝➜♣ ❝❛♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ tr tr✉♥❣ ❜➻♥❤ ✈➔ ♠ët sè ù♥❣ ❞ư♥❣ ❝õ❛ ❝❤ó♥❣ ✳ ✳ ✳ ✳ ✳ ✶✸✺ ❆✳✸✳ P❤➨♣ t➼♥❤ t➼❝❤ ♣❤➙♥ ❤➔♠ ♠ët ❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✸ ❆✳✸✳✶✳ ❚➼❝❤ ♣❤➙♥ ❜➜t ✤à♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✸ ▼Ö❈ ▲Ö❈ ✸ ❆✳✸✳✷✳ ❚➼❝❤ ♣❤➙♥ ①→❝ ✤à♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✼ ❆✳✸✳✸✳ ❚➼❝❤ ♣❤➙♥ s✉② rë♥❣ tr♦♥❣ tr÷í♥❣ ❤đ♣ ❝➟♥ ❧➜② t➼❝❤ ♣❤➙♥ ❧➔ ✈ỉ ❤↕♥ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✳ ✳ ✳ ✳ ✶✺✼ ✶✺✾ ❉❛♥❤ s→❝❤ ❤➻♥❤ ✈➩ ✶✳✶ ❱➼ ❞ö ✶✳✶✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✶✳✷ ❱➼ ❞ö ✶✳✶✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✶ ✣à♥❤ ♥❣❤➽❛ t➼❝❤ ♣❤➙♥ ❦➨♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✷ ❚➼❝❤ ♣❤➙♥ ♠✐➲♥ tê♥❣ q✉→t ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✸ ❚➼❝❤ ♣❤➙♥ ♠✐➲♥ tê♥❣ q✉→t ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✹ ✣ê✐ t❤ù tü t➼❝❤ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✺ ❱➼ ❞ö ✷✳✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✻ ❱➼ ❞ö ✷✳✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✼ ❱➼ ❞ö ✷✳✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✽ ❱➼ ❞ö ✷✳✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✾ ❱➼ ❞ö ✷✳✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✶✵ ▼✐➲♥ q✉↕t ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✶✶ ▼✐➲♥ q✉↕t ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✶✷ ▼✐➲♥ q✉↕t ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✶✸ ❱➼ ❞ö ✷✳✽ ❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✶✹ ❱➼ ❞ö ✷✳✽ ❜✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✶✻ ❱➼ ❞ö ✷✳✶✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✶✼ ❈❤ó þ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✶✽ ❱➼ ❞ö ✷✳✶✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✳✶✾ ❱➼ ❞ö ✷✳✶✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✳✷✵ ❉✐➺♥ t➼❝❤ ♠➦t ❝♦♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✷✳✷✶ ❱➼ ❞ö ✷✳✶✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✷✳✷✷ ❱➼ ❞ö ✷✳✶✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✷✳✷✸ ❱➼ ❞ö ✷✳✶✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✷✳✷✹ ❱➼ ❞ö ✷✳✶✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✷✳✷✺ ❱➼ ❞ö ✷✳✶✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✷✳✷✻ ❱➼ ❞ö ✷✳✶✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✷✳✷✼ ❱➼ ❞ö ✷✳✶✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✷✳✷✽ ✣à♥❤ ♥❣❤➽❛ t➼❝❤ ♣❤➙♥ ✤÷í♥❣ ❧♦↕✐ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✷✳✶✺ ❱➼ ❞ö ✷✳✾ ✷✳✷✾ ❱➼ ❞ö ✷✳✷✵ ❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✷✳✸✵ ❱➼ ❞ö ✷✳✷✵ ❜✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✷✳✸✶ ❱➼ ❞ö ✷✳✷✶ ❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✷✳✸✷ ❱➼ ❞ö ✷✳✷✶ ❜✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✷✳✸✸ ❈æ♥❣ t❤ù❝ ●r❡❡♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✷✳✸✹ ❈æ♥❣ t❤ù❝ ●r❡❡♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✷✳✸✺ ❈æ♥❣ t❤ù❝ ●r❡❡♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✷✳✸✻ ❈æ♥❣ t❤ù❝ ●r❡❡♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✷✳✸✼ ❱➼ ❞ö ✷✳✷✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✷✳✸✽ ❱➼ ❞ö ✷✳✷✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✷✳✸✾ ❚➼❝❤ ♣❤➙♥ ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✤÷í♥❣ ❧➜② t➼❝❤ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✷✳✹✵ ❚➼❝❤ ♣❤➙♥ ❦❤æ♥❣ ♣❤ư t❤✉ë❝ ✤÷í♥❣ ❧➜② t➼❝❤ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ❉❆◆❍ ❙⑩❈❍ ❍➐◆❍ ❱➇ ✺ ✷✳✹✶ ❚➼❝❤ ♣❤➙♥ ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✤÷í♥❣ ❧➜② t➼❝❤ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✷✳✹✷ ❍➺ q✉↔ ✷✳✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ❆✳✶ ❍➔♠ ❧÷đ♥❣ ❣✐→❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✹ ❆✳✷ ❍➔♠ ❛r❝t❛♥ ❆✳✸ ❍➔♠ ❛r❝❝♦t❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✺ ❆✳✹ ✣à♥❤ ♥❣❤➽❛ t➼❝❤ ♣❤➙♥ ①→❝ ✤à♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✺ ✻ ❉❆◆❍ ❙⑩❈❍ ❍➐◆❍ ❱➇ ❈⑩❈ ❑Þ ❍■➏❯ N✿ ❚➟♣ ❝→❝ sè tü ♥❤✐➯♥❀ N∗ ✿ ❚➟♣ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣❀ R ✿ ❚➟♣ ❝→❝ sè t❤ü❝❀ R∗ ✿ ❚➟♣ ❝→❝ sè t❤ü❝ ❦❤→❝ ✵❀ R∗+ ✿ ❚➟♣ ❝→❝ sè t❤ü❝ ❞÷ì♥❣❀ R+ ✿ ❚➟♣ ❝→❝ sè t❤ü❝ ❦❤æ♥❣ ➙♠❀ ∆ ✿ ❇➢t ✤➛✉ ❝❤ù♥❣ ♠✐♥❤❀ ✿ ❑➳t t❤ó❝ ❝❤ù♥❣ ♠✐♥❤✳ ✿ ✣à♥❤ ♥❣❤➽❛ ♦ ✿ ✣à♥❤ ỵ q ã ú ỵ ữỡ ❍⑨▼ ❙➮ ◆❍■➋❯ ❇■➌◆ ❙➮ ✶✳✶✳ ❑❤→✐ ♥✐➺♠ ♠ð ✤➛✉ ổ tr ỵ Rn x = (x1 , x2 , , xn )✱ ♠➔ t❛ ❝ô♥❣ ❣å✐ ❧➔ ❝→❝ ✤✐➸♠✳ x = (x1 , x2 , , xn ) ✈➔ y = (y1 , y2 , , yn ) ❝õ❛ Rn ❧➔ ❜✐➸✉ t❤ù❝ ❧➔ t➟♣ ❝→❝ ❜ë ❝â t❤ù tü ♥ sè t❤ü❝ ❚❛ ❣å✐ ❦❤♦↔♥❣ ❝→❝❤ ❣✐ú❛ ❤❛✐ ✤✐➸♠ ➮ (x1 − y1 )2 + (x2 − y2 )2 + + (xn − yn )2 d(x, y) = ❉➵ t❤➜② ❦❤♦↔♥❣ ❝→❝❤ tr♦♥❣ Rn ✤÷đ❝ ❝❤♦ ❜ð✐ ✭✶✳✶✮ ❝â ❜❛ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ s❛✉ ❝õ❛ ♠❡tr✐❝✿ ✭❛✮ d(x, y) ≥ 0, ∀x, y ∈ Rn , d(x, y) = ⇔ x = y; ✭❜✮ d(x, y) = d(y, x), ∀x, y ∈ Rn ; ✭❝✮ d(x, y) ≤ d(x, z) + d(z, y), ∀x, y, z ∈ Rn ◆❤÷ ✈➟② t Rn ợ ữủ ổ t❤ù❝ ✭✶✳✶✮ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ❬✷✱ tr ✸✾❪✳ ●✐↔ sû x∗ ∈ Rn ✈➔ ε > 0✳ ❚❛ ❣å✐ ε ✲ ❧➙♥ ❝➟♥ ❝õ❛ x∗ ❧➔ t➟♣ ❤ñ♣ s❛✉ ❝õ❛ Rn ✿ Vε (x∗ ) = {x ∈ Rn |d(x, x∗ ) < ε} ❚❛ ❣å✐ ❧➙♥ ❝➟♥ ❝õ❛ x x ữủ ỵ V n ∗ ❧➔ ♠å✐ t➟♣ ❝õ❛ R ❝❤ù❛ ✤÷đ❝ ♠ët ε ✲ ❧➙♥ ❝➟♥ ♥➔♦ ✤â ❝õ❛ x ✳ ▲➙♥ ❝➟♥ (x∗ )✳ ❚➟♣ Vε (x∗ ) = Vε (x∗ )\{x∗ } ✤÷đ❝ ❣å✐ ❧➔ ε✲ ❧➙♥ ❝➟♥ t❤õ♥❣ ❝õ❛ x∗ ✳ V (x ) = V (x )\{x } ✤÷đ❝ ❣å✐ ❧➔ ❧➙♥ ❝➟♥ t❤õ♥❣ ❝õ❛ x∗ ✳ n ∗ ●✐↔ sû D ⊂ R ✳ ✣✐➸♠ x ∈ D ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ tr♦♥❣ ❝õ❛ ❚➟♣ x tỗ t ởt ❧➙♥ ❝➟♥ ❝õ❛ ♥➡♠ ❤♦➔♥ t♦➔♥ tr♦♥❣ ❉ ✳ ❚➟♣ ❉ ✤÷đ❝ ❣å✐ ❧➔ ♠ð ♥➳✉ ♠å✐ ✤✐➸♠ ❝õ❛ ❉ ✤➲✉ ❧➔ ✤✐➸♠ tr♦♥❣ ❝õ❛ ♥â✳ ✣✐➸♠ y ∗ ∈ Rn ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ ❜✐➯♥ ❝õ❛ ❉ ♥➳✉ ♠å✐ ε✲ ❧➙♥ ❝➟♥ ❝õ❛ x∗ ✤➲✉ ✈ø❛ ❝❤ù❛ ✤✐➸♠ t❤✉ë❝ ❉✱ ✈ø❛ ❝❤ù❛ ✤✐➸♠ ❦❤æ♥❣ t❤✉ë❝ ❉✳ ✣✐➸♠ ❜✐➯♥ ❝õ❛ ❉ ❝â t❤➸ t❤✉ë❝ ❉✱ ❝ơ♥❣ ❝â t❤➸ ❦❤ỉ♥❣ t❤✉ë❝ ❉✳ ❚➟♣ ❝→❝ ✤✐➸♠ ❜✐➯♥ ❝õ❛ ❉ ✤÷đ❝ ❣å✐ ❧➔ õ ữủ ỵ D ❉ ✤÷đ❝ ❣å✐ ❧➔ ✤â♥❣ ♥➳✉ ♥â ❝❤ù❛ t➜t ❝↔ ❝→❝ ✤✐➸♠ ❜✐➯♥ ❝õ❛ ♥â✳ ∗ ∗ ❧➔ t➟♣ ♠ð✳ ❚❛ ❣å✐ Vε (x ) ❧➔ q✉↔ ❝➛✉ ♠ð t➙♠ x ✱ ❜→♥ ❦➼♥❤ ε✳ n ∗ n ∗ ❇✐➯♥ ❝õ❛ q✉↔ ❝➛✉ ➜② ❧➔ t➟♣ ❝→❝ ✤✐➸♠ x ∈ R s❛♦ ❝❤♦ d(x, x ) = ε ✳ ❚➟♣ {x ∈ R |d(x, x ) ≤ ε} ❱➼ ❞ö ε✲ ❧➙♥ ❝➟♥ Vε (x∗ ) ❝õ❛ x∗ ❧➔ ♠ët t➟♣ ✤â♥❣ ✈➔ ✤÷đ❝ ❣å✐ ❧➔ q✉↔ ❝➛✉ ✤â♥❣ t➙♠ x∗ ✱ ❜→♥ ❦➼♥❤ ε✳ ❍⑨▼ ❙➮ ◆❍■➋❯ ❇■➌◆ ❙➮ ữủ tỗ t↕✐ ♠ët q✉↔ ❝➛✉ ❝❤ù❛ ♥â✳ ❚➟♣ ❉ ✤÷đ❝ ❣å✐ ❧➔ ❧✐➯♥ t❤æ♥❣ ♥➳✉ ❝â t❤➸ ♥è✐ ❤❛✐ ✤✐➸♠ ❜➜t ❦ý ❝õ❛ ❉ ❜➡♥❣ ♠ët ✤÷í♥❣ ❧✐➯♥ tư❝ ♥➡♠ ❤♦➔♥ t♦➔♥ tr♦♥❣ ❉✳ ❚➟♣ ❉ ❧✐➯♥ t❤ỉ♥❣ ✤÷đ❝ ❣å✐ ❧➔ ỡ õ ỗ ởt t ữủ õ ỗ ♥❤✐➲✉ ♠➦t ❦➼♥ rí✐ ♥❤❛✉ tø♥❣ ✤ỉ✐ ♠ët✳ ✶✳✶✳✷✳ ✣à♥❤ ♥❣❤➽❛ ❤➔♠ sè ♥ ❜✐➳♥ sè ●✐↔ sû D ⊂ Rn ✳ ⑩♥❤ ①↕ f : D → R (x1 , x2 , xn ) → u = f (x1 , x2 , , xn ) ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ sè ♥ ❜✐➳♥ sè✳ ❚➟♣ ❉ ✤÷đ❝ ❣å✐ ❧➔ t➟♣ ①→❝ ✤à♥❤✱ x1 , x2 , , xn ✤÷đ❝ ❣å✐ ❧➔ ❝→❝ ❜✐➳♥ ✤ë❝ ❧➟♣✱ ✉ ✤÷đ❝ ❣å✐ ❧➔ ❜✐➳♥ tở tữớ ữủ ỵ ❤✐➺✉ ❧➔ z = f (x, y)✱ ❝á♥ ❤➔♠ ❜❛ tữớ ữủ ỵ u = f (x, y, z)✳ ❱➲ s❛✉ ♥❣♦➔✐ ❝→❝ ❝❤ú ❝→✐ ♥❤÷ x, y, z, t ỏ ỵ Rn ❜➡♥❣ ❝→❝ ❝❤ú ❝→✐ ✐♥ M, N, P, ✳ ụ ố ữ ợ ởt số ợ số t õ q ữợ ❜✐➳♥ sè ✤÷đ❝ ❝❤♦ ❜➡♥❣ ❜✐➸✉ t❤ù❝ ❣✐↔✐ t➼❝❤ u = f (x1 , x2 , , xn ) ✈➔ ❦❤ỉ♥❣ ❤♦❛ ♥❤÷ s❛✉✿ ♥â✐ ❣➻ t❤➯♠ ✈➲ t➟♣ ①→❝ số õ t t q ữợ t ①→❝ ✤à♥❤ ❝õ❛ ♥â ❧➔ t➟♣ t➜t ❝↔ n ❝→❝ ✤✐➸♠ M ∈ R ✱ s❛♦ ❝❤♦ f (M ) ❝â ♥❣❤➽❛✳ ❱➼ ❞ư ✶✳✶✳ • ❚➟♣ ①→❝ ✤à♥❤ ❝õ❛ ❤➔♠ z= − x2 − y ❧➔ t➟♣ ❝→❝ ✤✐➸♠ (x, y) ∈ R2 t❤♦↔ ♠➣♥ − x2 − y ≥ ⇔ x2 + y ≤ ✣â ❧➔ ❤➻♥❤ trá♥ t➙♠ ❖✭✵✱✵✮✱ ❜→♥ ❦➼♥❤ ❜➡♥❣ ✷✳ ✶✳✶✳✸✳ ●✐ỵ✐ ❤↕♥ ❝õ❛ ❤➔♠ ♥❤✐➲✉ ❜✐➳♥ ❈→❝ ❦❤→✐ ♥✐➺♠ tr♦♥❣ ♠ö❝ ♥➔②✱ ♠ö❝ ✶✳✶✳✹✱ ✈➔ tr♦♥❣ ❝→❝ ♣❤➛♥ ✶✳✷✱ ✶✳✸ ✤÷đ❝ tr➻♥❤ ❜➔② ❝❤♦ ❤➔♠ ❤❛✐ ❜✐➳♥✳ ❈❤ó♥❣ ❝â t❤➸ ✤÷đ❝ ♠ð rë♥❣ ❝❤♦ ❤➔♠ ♥❤✐➲✉ ❤ì♥ ❤❛✐ ❜✐➳♥✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ✈➔ ✈✐➳t Mn → M0 ❉➵ t❤➜② r➡♥❣ Mn (xn , yn ) ∈ R2 ✱ n ∈ N∗ ✱ ❞➛♥ ✤➳♥ ✤✐➸♠ M0 (x0 , y0 ) ∈ R2 ❝ü❝ ❤❛② Mn → M0 (n → ∞) ♥➳✉ d(Mn , M0 ) → 0(n → ∞)✳ ❚❛ ♥â✐ ❞➣② ✤✐➸♠ ❦❤✐ ♥ ❞➛♥ ✤➳♥ ✈æ Mn → M0 (n → ∞) ⇔ xn → x0 , yn → y0 (n → ∞) ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ●✐↔ sû ❤➔♠ z = f (x, y) M0 (x0 , y0 )✳ ❚❛ ♥â✐ ❤➔♠ f ❝â ❣✐ỵ✐ ❤↕♥ l lim f (x, y) = l ❤❛② lim f (M ) = l ♥➳✉ ✤✐➸♠ (x,y)→(x0 ,y0 ) Mn ∈ V (M0 ), ∀n M →M0 ∈ N ∗ , Mn → M0 (n → ∞) V (M0 ) ❝õ❛ ▼✭①✱②✮ ❞➛♥ ✤➳♥ M0 (x0 , y0 ) ✈➔ ✈✐➳t ♠å✐ ❞➣② ✤✐➸♠ Mn (xn , yn ) t❤ä❛ ♠➣♥ ①→❝ ✤à♥❤ tr♦♥❣ ❧➙♥ ❝➟♥ t❤õ♥❣ ❦❤✐ ✈ỵ✐ t❛ ✤➲✉ ❝â lim f (xn , yn ) = l n→∞ ✣à♥❤ ♥❣❤➽❛ ❤➔♠ ❝â ❣✐ỵ✐ ❤↕♥ ✈ỉ ❝ü❝ t÷ì♥❣ tü ♥❤÷ ✤à♥❤ ♥❣❤➽❛ tr➯♥✳ ◆❤➟♥ ①➨t ✶✳✶✳ ❈→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❤➔♠ sè ữ ợ tờ t tữỡ ỵ ❦➭♣✱✳✳✳ ✈➝♥ ❝á♥ ✤ó♥❣ ✈ỵ✐ ❣✐ỵ✐ ❤↕♥ ❝õ❛ ❤➔♠ ❤❛✐ ã Pì P tr♦♥❣ ✤â P✭①✱ ②✮✱ ◗✭①✱ ②✮ ❧➔ ❝→❝ ❤➔♠ ❧✐➯♥ tư❝ ❝ị♥❣ ❝→❝ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝➜♣ ♠ët ❝õ❛ ❝❤ó♥❣ tr♦♥❣ ♠ët ♠✐➯♥ ✤ì♥ ❧✐➯♥ ❉ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ∂Q ∂P = ∂x ∂y ✭✸✳✶✹✮ ✸✳✶✳✼✳✷✳ ❈→❝❤ ❣✐↔✐ ❚ø ✤✐➲✉ ❦✐➺♥ ✭✸✳✶✹✮ ♥➯♥ ✈➳ tr→✐ ♣❤÷ì♥❣ tr➻♥❤ ✭✸✳✶✸✮ ❧➔ ✈✐ ♣❤➙♥ t♦➔♥ ♣❤➛♥ ❝õ❛ ❤➔♠ ✉✭①✱②✮ ♥➔♦ ✤â✱ tù❝ ❧➔ ❞✉✭①✱ ②✮ ❂ P✭①✱ ②✮❞① ✰ ◗✭①✱ ②✮❞② ợ ữủ ởt tr ❝æ♥❣ t❤ù❝ s❛✉✿ y x P (x, y)dx + u(x, y) = x0 P (x, y0 )dx + u(x, y) = y0 y x ❤♦➦❝ Q(x0 , y)dy x0 Q(x, y)dy tr♦♥❣ ✤â ✭①0 ✱ ②0 ✮ ∈ ❉✳ y0 ❱➟② t➼❝❤ ♣❤➙♥ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t♦➔♥ ♣❤➛♥ ✭✸✳✶✸✮ ❧➔✿ y x P (x, y)dx + x0 y0 y x P (x, y0 )dx + ❤♦➦❝ Q(x0 , y)dy = C x0 Q(x, y)dy = C y0 ã ữỡ tr ✈✐ ♣❤➙♥ s❛✉ ❚❛ ❝â ❈→❝ ❤➔♠ P✱ ◗✱ (4xy + y)dx + (4x2 y + x)dy = P (x, y) = 4xy + y; Q(x, y) = 4x2 y + x ⇒ Py = 8xy + 1; Qx = 8xy + Py , Q x ❧✐➯♥ tư❝ ✈➔ Q x = Py ♥➯♥ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t♦➔♥ ♣❤➛♥✳ ❈❤å♥ ①0 ❂ ②0 ❂ ✵ t❛ ✤÷đ❝✿ y x u(x, y) = 0 ❱➟② t➼❝❤ ♣❤➙♥ tê♥❣ q✉→t ữỡ tr ú ỵ 0dy = 2x2 y + xy (4xy + y)dx + 2x2 y + xy = C ◆➳✉ P✭①✱ ②✮✱ ◗✭①✱ ②✮ ❧✐➯♥ tư❝ ❝ị♥❣ ✈ỵ✐ ❝→❝ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝➜♣ ✶ ❝õ❛ ❝❤ó♥❣ tr♦♥❣ ∂Q ∂P = ∂x ∂y ♠ët ♠✐➲♥ ❉ ♥➔♦ ✤â ♠➔ tr♦♥❣ ❉ t❤➻ P❞① ✰ ◗❞② ❂ ✵ ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t♦➔♥ ♣❤➛♥✳ ❚✉② ♥❤✐➯♥✱ tr♦♥❣ ♠ët sè tr÷í♥❣ ❤đ♣ ♥❣÷í✐ t❛ ❝â t❤➸ ❝❤å♥ ✤÷đ❝ ❤➔♠ ❤✭①✱ ②✮ ✤➸ ♣❤÷ì♥❣ tr➻♥❤ h(x, y) [P (x, y)dx + Q(x, y)dy] = ✭✸✳✶✺✮ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t♦➔♥ ♣❤➛♥✳ ❍➔♠ ❤✭①✱ ②✮ ❣å✐ ❧➔ t❤ø❛ sè t➼❝❤ ♣❤➙♥✳ ◆â✐ ❝❤✉♥❣✱ ❦❤æ♥❣ ❝â ♣❤÷ì♥❣ ♣❤→♣ tê♥❣ q✉→t ✤➸ t➻♠ t❤ø❛ sè t➼❝❤ ♣❤➙♥✳ ❚❛ ①➨t ✷ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t s❛✉✿ ❛✳ ◆➳✉ P y − Qx = φ(x) Q tù❝ ❧➔ ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ ② t❤➻ ❝❤å♥ ❤✭①✱ ②✮ ❂ ❤✭①✮ ❂ e ❚❤➟t ✈➟②✱ ✤➦t ❘✭①✱ ②✮ ❂ ❤✭①✱ ②✮ P✭①✱ ②✮❀ ❙✭①✱ ②✮ ❂ ❤✭①✱ ②✮ ◗✭①✱ ②✮✳ ❚❛ ❝â R = Pe S = Q.e φ(x)dx φ(x)dx ⇒ R y = Py e φ(x)dx ⇒ Sx = Qx e φ(x)dx + Q.φ(x).e φ(x)dx = [Qx + Qφ(x)] e φ(x)dx φ(x)dx ✸✳✷ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ✷ ✼✸ Py − Qx = φ(x) ⇒ Py = Qx + Q.φ(x)⇒ Sx = Py · e φ(x)dx Q ❱➟② Ry = Sx ♥➯♥ ✭✸✳✶✺✮ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t♦➔♥ ♣❤➛♥✳ Py − Qx ❜✳ ◆➳✉ = φ(y) tù❝ ❧➔ ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ ① t❤➻ ❝❤å♥ P ♠➔ ❤✭①✱ ②✮ ❂ ❤✭②✮ ❂ e− φ(y)dy ✭❝❤ù♥❣ ♠✐♥❤ tữỡ tỹ ữ tr trữớ ủ ã ✸✳✶✼✳ ▲í✐ ❣✐↔✐✳ 2 ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✭✷② ✰ ①② ✮ ❞① ✰ ✭①✰① ② ✮ ❞② ❂ ✵✳ ✣➦t P = 2y + xy , Py = + 3xy ❀ 2 ◗ ❂ ①✰① ② ✱ Qx ❂ ✶✰✷①② dx 2 x + x y Q ❚❛ ❝â Py − Qx = + xy = = ✳ ❱➟② t❤ø❛ sè t➼❝❤ ♣❤➙♥ ❤✭①✮ ❂ e x = eln |x| = x x x ◆❤➙♥ ✷ ✈➳ ữỡ tr ợ t ữủ ữỡ tr➻♥❤ ✈✐ ♣❤➙♥ t♦➔♥ ♣❤➛♥✿ x(2y + xy )dx + x(x + x2 y )dy = x ❈❤å♥ ①0 ❂ ②0 ❂ ✵ t❤➻ u(x, y) = x y (2xy +x2 y )dx+ 0 x2 x3 +y 3 0dy = 2y y = yx2 + x3 y 3 ❱➟② t➼❝❤ ♣❤➙♥ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❧➔ • ❱➼ ❞ư ✸✳✶✽✳ ▲í✐ ❣✐↔✐✳ ❚❛ ❝â ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✭x ✣➦t xy =C yx2 + + y )dx + (2xy + y x + ✱ P = x2 + y , Py = 2y Q = 2xy + y x + Py − Q x = −1 P ❤❛② ♥➯♥ t❤ø❛ sè t➼❝❤ ♣❤➙♥ ❤✭②✮ ❂ e 3yx2 + x3 y = C x3 )dy = x3 , Qx = 2y + y + x2 dy = ey y ◆❤➙♥ ✷ ✈➳ ❝õ❛ ữỡ tr ợ t ữủ ữỡ tr t♦➔♥ ♣❤➛♥✿ ey (x2 + y )dx + ey (2xy + y x + y x ❈❤å♥ ①0 ❂ ②0 ❂ ✵ t❤➻ x3 )dy = u (x, y) = (ey x2 + y ey )dx + 0dy = ey x3 + ey y x 3 ❱➟② t➼❝❤ ♣❤➙♥ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❧➔ ey x + ey y x = C ❤❛② ey (x3 + 3xy ) = C ✸✳✷✳ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ✷ ✸✳✷✳✶✳ ✣↕✐ ❝÷ì♥❣ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ✷ ✸✳✷✳✶✳✶✳ ✣à♥❤ ♥❣❤➽❛ ▲➔ ♣❤÷ì♥❣ tr➻♥❤ ❝â ❞↕♥❣✿ F (x, y, y , y ) = ✭✸✳✶✻✮ y = f (x, y, y ) ✭✸✳✶✼✮ ❤♦➦❝ tr♦♥❣ ✤â ②✿ ❤➔♠ ❝➛♥ t➻♠✱ ①✿ ❜✐➳♥ ✤ë❝ ❧➟♣✳ y, y ✿ ✤↕♦ ❤➔♠ ❝➜♣ ✶✱ ❝➜♣ ✷ ❝õ❛ ❤➔♠ ❝➛♥ t➻♠✳ ❇✐➳♥ ✤ë❝ ❧➟♣✱ ❤➔♠ ❝➛♥ t➻♠ ✈➔ ✤↕♦ ❤➔♠ ❝➜♣ ♠ët ❝â t❤➸ ❦❤ỉ♥❣ ❝â ♠ët ❝→❝❤ t÷í♥❣ ♠✐♥❤ ♥❤÷♥❣ ❜➢t ❜✉ë❝ ♣❤↔✐ ❝â ♠➦t ✤↕♦ ❤➔♠ ❝➜♣ ❤❛✐✳ P❍×❒◆● P ã ữỡ tr➻♥❤ ✿ y = 0, y y + (y )2 = 0, x2 y + xy + y = ❧➔ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ✷✳ ✸✳✷✳✶✳✷✳ ✣à♥❤ ỵ ỹ tỗ t t y = f (x, y, y )✳ ∂f (x, y, y ) ❧✐➯♥ tư❝ ∂y ❈❤♦ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ✷✿ ◆➳✉ f (x, y, y )✱ ∂f (x, y, y ) ∂y ✈➔ tr♦♥❣ ♠ët ♠✐➲♥ ❉ ♥➔♦ ✤â tr♦♥❣ ♥➳✉ ✭x0 , y0 , y0 ✮ ❧➔ ♠ët ✤✐➸♠ t❤✉ë❝ ❉ t❤➻ tr♦♥❣ ❧➙♥ ❝➟♥ ♥➔♦ ✤â ❝õ❛ ✤✐➸♠ x = x0 R3 tỗ t ởt ♥❤➜t ② ❂ ②✭①✮ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✸✳✶✼✮ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥✿ y|x=x0 = y0 , y |x=x0 = y1 ✭✸✳✶✽✮ ❇➔✐ t♦→♥ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✸✳✶✼✮ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭✸✳✶✽✮ ✤÷đ❝ ❣å✐ ❧➔ ❜➔✐ t♦→♥ ❈❛✉❝❤②✳ ✸✳✷✳✶✳✸✳ ◆❣❤✐➺♠ tê♥❣ q✉→t✱ ♥❣❤✐➺♠ r✐➯♥❣✱ t➼❝❤ ♣❤➙♥ tê♥❣ q✉→t✱ t➼❝❤ ♣❤➙♥ r✐➯♥❣ ✲ ◆❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ✷ ❧➔ ❤➔♠ sè ② ❂ ϕ ✭①✱ ❈1 ✱ ❈2 ✮ tr♦♥❣ ✤â ❈1 ✱ ❈2 ✿ ❤➡♥❣ sè✳ ✲ ◆❣❤✐➺♠ ♥❤➟♥ ✤÷đ❝ tø ♥❣❤✐➺♠ tê♥❣ q✉→t ❜➡♥❣ ❝→❝❤ ❝❤♦ ❈1 ❂ C1 ✱ ❈2 ❂ = ϕ(x, C10 , C20 ) t❤ä❛ ♠➣♥ y|x=x0 = y0 , y |x=x0 = y1 ❣å✐ ❧➔ ♥❣❤✐➺♠ r✐➯♥❣✳ y ✲ ❍➺ t❤ù❝ φ C20 s❛♦ ❝❤♦ ❤➔♠ sè ✭①✱②✱❈1 ✱❈2 ✮ ❂ ✵ ①→❝ tờ qt ữỡ tr ữợ ➞♥ ❣å✐ ❧➔ t➼❝❤ ♣❤➙♥ tê♥❣ q✉→t✳ ✲ ❑❤✐ ❝❤♦ ❈1 ✱ ❈2 ❝→❝ ❣✐→ trà ❝ö t❤➸ C1 = C1 , C2 0 ❤➺ t❤ù❝ φ(x, y, C1 , C2 ✮❂ ✵ ❣å✐ ❧➔ t➼❝❤ ♣❤➙♥ r✐➯♥❣✳ = C20 tø t➼❝❤ ♣❤➙♥ tê♥❣ q✉→t t❛ ♥❤➟♥ ✤÷đ❝ ✸✳✷✳✷✳ P❤÷ì♥❣ tr➻♥❤ ❦❤✉②➳t ✸✳✷✳✷✳✶✳ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ✷ ❦❤✉②➳t y, y ❛✳ ❉↕♥❣✿ F (x, y ) = ❜✳ ❈→❝❤ ❣✐↔✐ y =p y = f (x) ✭✸✳✶✾✮ ✳ ▲➜② t➼❝❤ ♣❤➙♥ t❤❡♦ ❜✐➳♥ ① ❤❛✐ ❧➛♥ t❛ ✤÷đ❝ ❝ỉ♥❣ t❤ù❝ ♥❣❤✐➺♠ tê♥❣ q✉→t✿ ⑩ y= ❤♦➦❝ ✤➦t ❤❛② ❿ f (x)dx dx + C1 x + C2 t❛ ✤÷đ❝ ❋✭x, p ✮ ❂ ✵ ❧➔ ữỡ tr ởt ố ợ tờ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤â ❧➔ ♣ ❂ ❢✭①✱ ❈1 ✮ t❤➻ ② ❂ ❣✭①✱ ❈1 ✮ ✰ ❈2 tr♦♥❣ ✤â ❣✭①✱ ❈1 ✮ ❧➔ ♥❣✉②➯♥ ❤➔♠ ❝õ❛ ❢✭①✱ ❈1 ã ữỡ tr ✈✐ ♣❤➙♥ y = sin x cos2 x − sin3 x ▲➜② t➼❝❤ ♣❤➙♥ t❤❡♦ ❜✐➳♥ ① ❤❛✐ ❧➛♥ t❛ ✤÷đ❝✿ ➌ ➉ sin x cos x − sin x dx dx + C1 x + C2 y= ➉ = ➌ 2 −2cos x + − cos x d cos x dx + C1 x + C2 ✸✳✷ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ✷ ✼✺ (cos x − cos3 x)dx + C1 x + C2 = (cos x − = cos(3x) + 3cosx )dx + C1 x + C2 − cos(3x) + cosx )dx + C1 x + C2 −sin3x sinx + + C1 x + C2 = 12 = ( ✸✳✷✳✷✳✷✳ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ✷ ❦❤✉②➳t ② ❛✳ ❉↕♥❣✿ F (x, y , y ) = ❜✳ ❈→❝❤ ❣✐↔✐ ❑❤✐ ✤â ✳ ✣➦t ♣ ❂ y =p ❱➼ ❞ư ✸✳✷✶✳ ▲í✐ ❣✐↔✐✳ y = f (x, y ) ✭✸✳✷✵✮ ✭ ♣ ❧➔ ❤➔♠ ❝õ❛ ①✮✳ p ❂ ❢✭①✱ ♣✮ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ✶ ✤è✐ ϕ ✭①✱ ❈1 ✮✳ ❉♦ ✤â ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ✭✸✳✷✵✮ ❧➔ ②❂ φ(x, C1 )dx+C2 ✈➔ ♣❤÷ì♥❣ tr tr t ợ r t ữủ ã y ữỡ tr t y = y =p⇒y =p y + x x t❤❛② ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ t❛ ✤÷đ❝ p − p =x x ✭✯✮ ✭✯✮ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ❝➜♣ ✶ ố ợ ữỡ tr p dp dx = ❤❛② = ⇒ p = Cx ✭❈✿❝♦♥st✮✳ x p x ✰✮ ❈♦✐ ❈❂❈✭①✮ t❤➻ p = C x + C ✳ ❚❤❛② p, p ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✯✮ t❛ ✤÷đ❝ C x + C − C = x ❤❛② C = 1⇒ C = x + C1 ✭C1 ✿ ❝♦♥st✮✳ ❉♦ ✤â ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✯✮ ❧➔ p = (x + C1 )x ✭C1 ✿ ❝♦♥st✮✳ x3 x2 ❱➟② ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❧➔ y = x(C1 + x)dx = + C1 + C2 ✰✮ P❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t p − ✸✳✷✳✷✳✸✳ P❤÷ì♥❣ tr➻♥❤ ❦❤✉②➳t ① ❛✳ ❉↕♥❣✿ F (y, y , y ) = ❜✳ ❈→❝❤ ❣✐↔✐ ❑❤✐ ✤â ✳ ✣➦t y ❤❛② y = f (y, y ) ❂ ♣ tr♦♥❣ ✤â ♣ ❧➔ ❤➔♠ ❝õ❛ ② tù❝ ❧➔ dp y = py yx = p ✳ dy ✭✸✳✷✶✮ p = p [y(x)] ❚❤❛② ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✸✳✷✶✮ t❛ ✤÷đ❝✿ p dp = f (y, p) dy ✭✸✳✷✷✮ ✭✸✳✷✷✮ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ♠ët ♠➔ ❤➔♠ ❝➛♥ t➻♠ ❧➔ ♣✱ ❝á♥ ❜✐➳♥ ❧➔ ②✳ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② t❛ ❝â ♥❣❤✐➺♠ p = φ(y, C1 ) ❤❛② dy = dx φ(y, C1 ) ❉♦ ✤â t➼❝❤ ♣❤➙♥ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✸✳✷✶✮ ❧➔ ❱➼ ❞ư ✸✳✷✷✳ • 1, y |x=0 = ❚➻♠ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ dy = x + C2 ϕ(y, C1 ) yy − (y )2 = t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ y|x=0 = P❍×❒◆● ❚❘➐◆❍ ❱■ P❍❹◆ ✼✻ ▲í✐ ❣✐↔✐✳ dp y.p = p2 dy ✣➦t ❤❛② y = p = p [y(x)] ⇒ y = py y x = ppy y.p.dp = p2 dy ✳ t❤❛② ữỡ tr t ữủ ợ ② ❂ ✵ t❤û trü❝ t✐➳♣ ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ❧➔ ♥❣❤✐➺♠✳ ⇒y ❂ ✵ ⇒ ② ❂ ❈ t❤❛② ✈➔♦ y · p = ❝❤✐❛ ❝↔ ✷ ✈➳ ❝õ❛ ữỡ ợ ợ ữỡ tr➻♥❤ ❧➔ ♥❣❤✐➺♠✳ tr➻♥❤ ✭✯✮ ❝❤♦ y · p2 t❛ ✤÷đ❝✿ dp dy = ⇒ p = C1 y ⇒ y = C1 y p y hay dy dy = C1 y ⇒ = C1 dx dx y ⇒ ln |y| = C1 x + ln |C2 | hay y = C2 eC1 x ➜ ❚❤❛② ✤✐➲✉ ❦✐➺♥ ✤➛✉ y |x=0 = 2, y|x=0 = t❛ ✤÷đ❝ ❱➟② ♥❣❤✐➺♠ r✐➯♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❧➔ ② ❂ C1 = C2 = e2x ✸✳✷✳✸✳ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ❤❛✐ ❝â ❤➺ sè t❤❛② ✤ê✐ ✸✳✷✳✸✳✶✳ ✣à♥❤ ♥❣❤➽❛ ▲➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝â ❞↕♥❣✿ y + p(x)y + q(x).y = f (x) p(x), q(x), f (x) ❧➔ ♥❤ú♥❣ ❤➔♠ sè ❧✐➯♥ tö❝✳ f (x) ≡ t❤➻ ✭✸✳✷✸✮ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t✳ f (x) = t❤➻ ✭✸✳✷✸✮ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❦❤ỉ♥❣ t❤✉➛♥ ✭✸✳✷✸✮ tr♦♥❣ ✤â ✲ ◆➳✉ ✲ ◆➳✉ ♥❤➜t✳ ✸✳✷✳✸✳✷✳ P❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ❝➜♣ ✷ t❤✉➛♥ ♥❤➜t ❛✳ ❉↕♥❣✿ y + p(x).y + q(x).y = ỵ t ỵ y (x), y (x) ❧➔ ❤❛✐ ♥❣❤✐➺♠ ❝õ❛ ✭✸✳✷✹✮ t❤➻ y = C1y1(x) + C2y2(x) ✭C1, C2✿ ❤➡♥❣ sè✮ ❝ô♥❣ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t ✭✸✳✷✹✮✳ ✣➦❝ ❜✐➺t y1 (x) ✈➔ y2 (x) y = C1 y1 (x) + C2 y2 (x) ♦ ✭✸✳✷✹✮ ❧➔ ❤❛✐ ♥❣❤✐➺♠ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ❝õ❛ ✭✸✳✷✹✮ ✭tù❝ ❧➔ y1 (x) = C✮ y2 (x) t❤➻ ❧➔ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ữỡ tr t t ỵ t ♠ët ♥❣❤✐➺♠ r✐➯♥❣ y (x) ✭✈ỵ✐ y (x) = 0✮ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t 1 t❤➻ t❛ ❝â t❤➸ t➻♠ ✤÷đ❝ ♥❣❤✐➺♠ r✐➯♥❣ t❤ù ❤❛✐ ②2✭①✮ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t ✭✸✳✷✹✮ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ✈ỵ✐ ②1✭①✮ t ú ỵ ✣➸ t➻♠ ♥❣❤✐➺♠ r✐➯♥❣ t❤ù ✷ t❛ ❝â t❤➸ sû ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ ▲✐♦✉✈✐❧❧❡✿ y2 (x) = y1 (x) e− p(x)dx dx y12 (x) ✸✳✷ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ✷ ❱➼ ❞ư ✸✳✷✸✳ • y1 (x) = x ▲í✐ ❣✐↔✐✳ ✼✼ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✣➦t (1 − x2 )y − 2xy + 2y = ❜✐➳t ♠ët ♥❣❤✐➺♠ r✐➯♥❣ ❝õ❛ ♥â ❧➔ y2 (x) = u.x ⇒ y2 (x) = u + xu ✱ y2 (x) = 2u + xu t❤❛② ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ t❛ ✤÷đ❝ (1 − x2 )(2u + xu ) − 2x(u + xu ) + 2xu = hay (x − x3 )u + (2 − 4x2 )u = ✣➦t u❂ ♣✱ ♣❤÷ì♥❣ tr➻♥❤ trð t❤➔♥❤ ❉♦ ✤â p= 1 − dx = ln x2 |1 − x2 | − + x 1−x 1+x u ❈❤å♥ ❑ ❂ ✵ t❛ ✤÷đ❝ ⑩ ◆❤➟♥ ①➨t ✸✳✶✳ ❿ 1 1+x u(x) = − + ln x 1−x 1+x y2 (x) = u(x) · x = −1 + x ln 1−x 1+x y = C1 x + C2 x ln − C2 1−x ✳ ❉♦ ✤â y2 (x) t❛ ❝â t❤➸ sû ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ ▲✐♦✉✈✐❧❧❡ ✈➔ ❝â ❦➳t q✉↔ ✣➸ t➻♠ ♥❣❤✐➺♠ r✐➯♥❣ 1+x y2 (x) = −1 + x ln 1−x ❱➼ ❞ư ✸✳✷✹✳ ▲í✐ ❣✐↔✐✳ (2x − x2 )y + (x2 − 2)y + 2(1 − x)y = ❝â ♠ët ♥❣❤✐➺♠ y = ex ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ ❜✐➳t y|x=1 = 0, y |x=1 = 1✳ ❈❤♦ ♣❤÷ì♥❣ tr➻♥❤ ❍➣② t➻♠ ♥❣❤✐➺♠ r✐➯♥❣ ❝õ❛ ✣➦t y(x) = ex ✳ ❚❛ t➻♠ ②2 ✭①✮ ✤ë❝ ❧➟♣ ✈ỵ✐ ②1 ✭①✮ t❤❡♦ ❝ỉ♥❣ t❤ù❝ ▲✐♦✉✈✐❧❧❡✿ x2 − dx 2x − x2 − x y2 (x) = e − x2 − dx = 2x − x2 ❈❤♦ ❈ ❂ ✵ t❛ ✤÷đ❝ = ex ⑩ e e2x ❿ (x2 − 2x)e−x dx = ex −(x2 − 2x)e−x − 2xe−x = −x2 ❚❤❛② ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ ✈➔♦ ② ✈➔ ❱➟② ♥❣❤✐➺♠ r✐➯♥❣ ❝➛♥ t➻♠ ❧➔ ❱➼ ❞ö ✸✳✷✺✳ dx 1 1+ + dx = x + ln |x(x − 2)| + C x x−2 ex x(x − 2) y2 (x) = ex dx e2x ❱➟② ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❧➔ • + ln |C| 1+x 1 + K dx = − + ln + x2 − x2 x 1−x ❱➟② ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❧➔ ❚➼♥❤ −1 t❛ ❝â dx ⇒ u(x) = du = x (1 − x2 ) • dp 4x2 − = dx p x − x3 C x2 (1 − x2 ) ❈❤å♥ ❈ ❂ ✶✱ t❤❛② ♣ ❂ t÷ì♥❣ tü ❤❛② ❿ ⑩ 4x2 − dx = x − x3 ⇒ ln |p| = (x − x3 )dp = (4x2 − 2)pdx y y = C e x − C x2 ✳ t❛ ✤÷đ❝ y = C1 ex − 2C2 x ➜ C1 e − C2 = C1 = − hay e C1 e − 2C2 = C2 = −1 y = −ex−1 + x2 ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ (x2 − 1)y + 2xy = õ Pì P Pữỡ tr➻♥❤ ✤➣ ❝❤♦ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ✷ ❦❤✉②➳t ②✳ ❱➟② t❛ ❝â t❤➸ ❣✐↔✐ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ỗ tớ õ ụ ởt ữỡ tr ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ✷ t❤✉➛♥ ♥❤➜t ❝â ❤➺ sè t❤❛② ✤ê✐ ♥➯♥ ❝â t❤➸ ❣✐↔✐ ❜➡♥❣ ❝→❝❤ t➻♠ ❤❛✐ ♥❣❤✐➺♠ r✐➯♥❣ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✳ ❈→❝❤ ✶ ✳ ✣➦t y =p t❤➻ (x2 − 1)p + 2xp = y =p ❤❛② t❤❛② ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ t❛ ✤÷đ❝✿ 2x − dx dp 2xdx 2x p=0⇒ =− hay p = C1 e x2 − = p + x −1 p x −1 C1 −1 x2 ❚❤❛② ♣ ❂y t❛ ✤÷đ❝ y = C1 hay y = x2 − ❱➟② ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❈→❝❤ ✷✳ x−1 C1 + C2 dx = C ln x2 − x+1 x−1 ❧➔ y = C1 ln + C2 x+1 ❇➡♥❣ ❝→❝❤ t❤❛② ② ❂ ❈ ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ t❤➻ ② ❂ ❈ ❧➔ ♥❣❤✐➺♠✳ ❈❤å♥ ❈ ❂ ✶ t❛ ✤÷đ❝ ② ❂ ✶ ❧➔ ♠ët ♥❣❤✐➺♠ r✐➯♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦✳ ❚➻♠ ♥❣❤✐➺♠ r✐➯♥❣ t❤ù ✷ t❤❡♦ ❝æ♥❣ t❤ù❝ 2x dx − dx = ln x − + K ▲✐♦✉✈✐❧❧❡ y2 (x) = e x+1 x−1 ❈❤å♥ ❑ ❂ ✵ ⇒ y2 (x) = ln x+1 − x2 ❱➟② ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❧➔ y = C1 + C2 ln x−1 x+1 ✸✳✷✳✸✳✸✳ P❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ❝➜♣ ❤❛✐ ❦❤ỉ♥❣ t❤✉➛♥ ♥❤➜t ❛✳ P❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ❝➜♣ ❤❛✐ ❦❤ỉ♥❣ t❤✉➛♥ ♥❤➜t ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❝â ❞↕♥❣✿ y + p(x).y + q(x).y = f (x) tr♦♥❣ ✤â ✭✸✳✷✺✮ f (x) = 0✳ ❜✳ ỵ t ỵ ✸✳✸✳ ◆❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❦❤ỉ♥❣ t❤✉➛♥ ♥❤➜t ✭✸✳✷✺✮ ❜➡♥❣ tê♥❣ ❝õ❛ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t t tữỡ ự ợ ởt r õ ữỡ tr ổ t t ỵ y = y + Y ỵ ỵ ỗ t ữỡ tr y + p(x)y + q(x)y = f1 (x) + · · · + fn (x) ✭✸✳✷✻✮ ◆➳✉ ②i✭①✮ (i = 1, n) ❧➔ ♥❣❤✐➺♠ r✐➯♥❣ y = y1 (x) + y2 (x) + · · · + yn (x) ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ y + p(x)y + q(x)y = fi(x) t❤➻ r ữỡ tr ỵ ②1 ✭①✮ ✈➔ ②2 ✭①✮ ❧➔ ❤❛✐ ♥❣❤✐➺♠ r✐➯♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❦❤ỉ♥❣ t❤✉➛♥ ♥❤➜t ✭✸✳✷✺✮ t❤➻ ②✭①✮ ❂ ②1 ✭①✮−②2 ✭①✮ ❧➔ ♥❣❤✐➺♠ r✐➯♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t ✭✸✳✷✹✮✳ ❝✳ P❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ t❤✐➯♥ ❤➡♥❣ sè ▲❛❣r❛♥❣❡ ◆➳✉ ❜✐➳t y = C1 y1 + C2 y2 ✭❈1 ✱ ❈2 ✿ ❝♦♥st✮ ❧➔ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t ✭✸✳✷✹✮ t❤➻ ❝â t❤➸ t➻♠ ♥❣❤✐➺♠ tê♥❣ q✉→t ữỡ tr ữợ t ❈2 ✭①✮ t❛ ❣✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✿ ➜ C1 y1 + C2 y2 = C1 y1 + C2 y2 = f (x) y = C1 (x)y1 +C2 (x)y2 ✳ ✸✳✷ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ✷ ✼✾ C1 = ϕ1 (x), C2 = ϕ2 (x)✳ ❚ø ✤â s✉② r❛✿ C1 = ϕ1 (x)dx = Φ1 (x) + K1 ; C2 = ϕ2 (x)dx = Φ2 (x) + K2 ✭❑1 ✱ ❑2 ✿ ❝♦♥st✮✳ ❱➟② ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ✭✸✳✷✺✮ ❧➔ y = K1 y1 + K2 y2 + Φ1 (x)y1 + Φ2 (x)y2 ✭❑1 ✱ ❑❤✐ ✤â • ❱➼ ❞ư ✸✳✷✻✳ ▲í✐ ❣✐↔✐✳ ❚➻♠ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ y − ❑2 ✿ ❝♦♥st✮✳ y = x x ❳➨t ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ t❤✉➛♥ ♥❤➜t t÷ì♥❣ ù♥❣ y − y = x P❤÷ì♥❣ tr➻♥❤ ❦❤ỉ♥❣ ❝❤ù❛ ❤➔♠ ②✱ ❞♦ ✤â ♥â ❝â ♠ët ♥❣❤✐➺♠ r✐➯♥❣ ②1 ❂ ✶✳ ◆❣❤✐➺♠ r✐➯♥❣ ②2 t➻♠ ❜➡♥❣ ❝æ♥❣ t❤ù❝ ▲✐♦✉✈✐❧❧❡ y2 (x) = ❱➟② ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t ❧➔ x2 e− (− x )dx dx = 2 y = C1 x + C2 ✭❈1 ✱❈2 ✿ ❝♦♥st✮✳ ❚❛ t➻♠ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❦❤ỉ♥❣ t❤✉➛♥ ♥❤➜t ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ t❤✐➯♥ ❤➡♥❣ sè ▲❛❣r❛♥❣❡✳ ◆❣❤✐➺♠ tê♥❣ q✉→t ❝â ❞↕♥❣ y = C1 (x)x + C2 (x) tr♦♥❣ ✤â C1 (x), C2 (x) ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❤➺✿ ➜ ❱➟② ♥❣❤✐➺♠ tê♥❣ q✉→t ữỡ tr ã x C1 (x) = + K1 ⇒  C (x) = − x + K 2 x3 x3 x3 y= − + K x2 + K = + K x2 + K    C1 (x) = C1 (x) · x2 + C2 (x) · = 2 ❤❛② C1 (x) · 2x + C2 (x) · = x   C2 (x) = − x ❱➼ ❞ư ✸✳✷✼✳ tr➻♥❤ ❧➔ y1∗ = ▲í✐ ❣✐↔✐✳ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ (x x2 + x + x, y2∗ = − 1)y + 4xy + 2y = 6x✳ ❇✐➳t ❤❛✐ ♥❣❤✐➺♠ r✐➯♥❣ ❝õ❛ ♣❤÷ì♥❣ x+1 ❱➻ y1∗ , y2∗ ❧➔ ❤❛✐ ♥❣❤✐➺♠ r✐➯♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ♥➯♥ ❧➔ ♠ët ♥❣❤✐➺♠ r✐➯♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t t÷ì♥❣ ù♥❣ y1 = y2∗ − y1∗ = x+1 (x2 − 1)y + 4xy + 2y = 0✳ ✭✯✮ ◆❣❤✐➺♠ r✐➯♥❣ ②2 ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✯✮ ✤÷đ❝ t➻♠ ❜ð✐ ❝ỉ♥❣ t❤ù❝ ▲✐♦✉✈✐❧❧❡✿ 4x dx −1 e −1 dx = x −1 (x + 1)2 − y2 = x+1 x2 ❉♦ ✤â ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t t÷ì♥❣ ù♥❣ ❧➔ y = C1 1 +C2 x+1 x −1 ❱➟② ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❧➔ y = C1 1 + C2 + (C1 , C2 : const) x+1 x −1 x+1 ✸✳✷✳✹✳ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ❤❛✐ ❝â ❤➺ sè ❦❤ỉ♥❣ ✤ê✐ ✸✳✷✳✹✳✶✳ P❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t ❛✳ ❉↕♥❣ ♣❤÷ì♥❣ tr➻♥❤✿ y + py + qy = tr♦♥❣ ✤â p, q ❜✳ ❈→❝❤ ❣✐↔✐ ❧➔ ❝→❝ ❤➡♥❣ sè t❤ü❝✳ ✭✸✳✷✼✮ P❍×❒◆● ❚❘➐◆❍ ❱■ P❍❹◆ ✽✵ kx ❚➻♠ r õ ữợ y = e tr ✤â ❦ ❧➔ ❤➡♥❣ sè✳ ❚❛ ❝â kx ❚❤❛② y , y ✈➔♦ ✭✸✳✷✼✮ t❛ ✤÷đ❝ e (k + pk + q) = 0✳ ❙✉② r❛ k + pk + q = y = kekx , y = k ekx ✳ ✭✸✳✷✽✮ ✭✸✳✷✽✮ ✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❝õ❛ ✭✸✳✷✼✮✳ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ k + pk + q = ✲ ◆➳✉ ♣❤÷ì♥❣ tr➻♥❤ ✭✸✳✷✽✮ ❝â ✷ ♥❣❤✐➺♠ t❤ü❝ ❦1 k x k x ✭✸✳✷✼✮ ❧➔ y = C1 e + C2 e ✭C1 , C2 : const✮✳ = ❦2 t❤➻ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✲ ◆➳✉ ♣❤÷ì♥❣ tr➻♥❤ ✭✸✳✷✽✮ ❝â ♥❣❤✐➺♠ ❦➨♣ ❦1 ❂ ❦2 ❂ ❦ t❤➻ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ kx tr➻♥❤ ✭✸✳✷✼✮ ❧➔ y = C1 e + C2 xekx ❂(C1 + C2 x)ekx ✭C1 , C2 : const✮✳ ✲ ◆➳✉ ♣❤÷ì♥❣ tr➻♥❤ ✭✸✳✷✽✮ ❝â ♥❣❤✐➺♠ ♣❤ù❝ ❦ ❂ α αx tr➻♥❤ ✭✸✳✷✼✮ ❧➔ y = e (C1 cos βx + C2 sin βx) ✭C1 , C2 : ❱➼ ❞ư ✸✳✷✽✳ • ± ✐β t❤➻ const✮✳ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ ●✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ s❛✉✿ y − 5y + 6y = 0❀ ❜✮ y − 2y + 3y = 0❀ ❝✮ y − 4y + 4y = 0✳ ❛✮ ▲í✐ ❣✐↔✐✳ k − 5k + = ⇒ k1 = 2, k2 = 2x ❱➟② ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❧➔ y = C1 e + C2 e3x (C1 , C2 : const) √ ❜✮ P❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❧➔ k − 2k + = 0⇒ k = ± i √ √ x ❱➟② ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❧➔ y = e (C1 cos 2x + C2 sin 2x) ❝✮ P❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❧➔ k − 4k + = ⇒ k = 2x + C2 xe2x ❱➟② ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❧➔ y = C1 e ❛✮ P❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❧➔ ✸✳✷✳✹✳✷✳ P❤÷ì♥❣ tr➻♥❤ ❦❤ỉ♥❣ t❤✉➛♥ ♥❤➜t ❛✳ ❉↕♥❣ ♣❤÷ì♥❣ tr➻♥❤✿ y + py + qy = f (x) tr♦♥❣ ✤â ♣✱ q ❧➔ ❤➡♥❣ sè✱ ❢✭①✮ ❜✳ ❈→❝❤ ❣✐↔✐ ❝❤✉♥❣ = ✭✸✳✷✾✮ ✵✳ ✣➸ ❣✐↔✐ ✭✸✳✷✾✮ t❛ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t t÷ì♥❣ ù♥❣ ✭✸✳✷✼✮ s❛✉ ✤â ❞ị♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ t❤✐➯♥ ❤➡♥❣ sè ▲❛❣r❛♥❣❡✳ ◆❤÷♥❣ tr♦♥❣ ♠ët sè tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ữợ õ t t ữủ ởt ♥❣❤✐➺♠ r✐➯♥❣ ❝õ❛ ✭✸✳✷✾✮ ❜➡♥❣ ❝→❝❤ ❣✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè✳ ❑❤✐ ✤â ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✸✳✷✾✮ ❜➡♥❣ tê♥❣ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ t tữỡ ự ợ ởt r rữớ ủ ợ α f (x) = eαx Pn (x) tr♦♥❣ ✤â α ❧➔ ❤➡♥❣ sè✱ Pn ✭①✮ ❧➔ ✤❛ t❤ù❝ ❜➟❝ ♥ ❝õ❛ ❦❤ỉ♥❣ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ✭✸✳✷✽✮ t❤➻ t❛ t➻♠ ♥❣❤✐➺♠ r✐➯♥❣ ❝õ❛ Y = eαx Qn (x) tr♦♥❣ ✤â ◗n ✭①✮ ❧➔ ✤❛ t❤ù❝ ❝ò♥❣ ❜➟❝ ợ Pn ữợ ✤ì♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ✭✸✳✷✽✮ t❤➻ t❛ t➻♠ ♥❣❤✐➺♠ r✐➯♥❣ ❝õ❛ ✭✸✳✷✾✮ Y = xeα x Qn (x) tr♦♥❣ ✤â ◗n ✭①✮ ❧➔ ✤❛ t❤ù❝ ❝ò♥❣ ❜➟❝ ♥ ✈ỵ✐ Pn ✭①✮✳ ✲ ◆➳✉ α ❧➔ ♥❣❤✐➺♠ ❦➨♣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ✭✸✳✷✽✮ t❤➻ t❛ t➻♠ ♥❣❤✐➺♠ r✐➯♥❣ x ữợ Y = x e Qn (x) tr♦♥❣ ✤â ◗n ✭①✮ ❧➔ ✤❛ t❤ù❝ ❝ò♥❣ ợ Pn ữợ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ✷ ✯ ❚r÷í♥❣ ❤đ♣ ✷ ✳ ❱ỵ✐ ✽✶ f (x) = eαx [Pn (x) cos βx + Qm (x) sin βx] tr♦♥❣ ✤â α✱ β ❧➔ ❤➡♥❣ sè❀ Pn ✭①✮✱ ◗m ✭①✮ ❧➛♥ ❧÷đt ❧➔ ❝→❝ ✤❛ t❤ù❝ ❜➟❝ ♥✱ ♠✳ ✲ ◆➳✉ α ± ✐β ❦❤ỉ♥❣ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ✭✸✳✷✽✮ t❤➻ t❛ t➻♠ ♥❣❤✐➺♠ r✐➯♥❣ Y = eαx (Rl (x) cos βx + Hl (x) sin βx) tr♦♥❣ ✤â Rl (x), Hl (x) tự ữợ ❞↕♥❣ l, l = max(n, m)✳ ✲ ◆➳✉ α ± ✐β ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ✭✸✳✷✽✮ t❤➻ t t r x ữợ Y = xe [Rl (x) cos βx + Hl (x) sin βx] tr♦♥❣ ✤â Rl (x), Hl (x) ❧➔ ❝→❝ ✤❛ t❤ù❝ ❜➟❝ l, l = max(n, m)✳ ❜➟❝ • ❱➼ ❞ư ✸✳✷✾✳ ●✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✿ y − 4y + 3y = e4x ❀ x ❜✮ y + 4y − 5y = e (x + 1)❀ 3x ❝✮ y − 6y + 9y = e ✳ ❛✮ ▲í✐ ữợ Pữỡ tr t t tữỡ ù♥❣ y − 4y + 3y = ❝â ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣✿ k − 4k + = ⇒ k1 = 1; k2 = ❉♦ ✤â ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t t÷ì♥❣ ù♥❣ ữợ ữỡ tr ✤➣ ❝❤♦ ❧➔ f (x) = e4x , α = y = C1 ex + C2 e3x ❦❤æ♥❣ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ♥➯♥ t❛ t➻♠ ởt r ữỡ tr ữợ Y = A.e4x ⇒ Y = 4A.e4x , Y = 16.A.e4x ❚❤❛② Y, Y , Y ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ t❛ ✤÷đ❝✿ 16Ae4x − 4.4Ae4x + 3Ae4x = e4x ⇒ 3A = ↔ A = Y = e4x ❱➟② ♥❣❤✐➺♠ r✐➯♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❧➔ y = C1 ex + C2 e3x + e4x t÷ì♥❣ ù♥❣ y + 4y − 5y = ❝â ❉♦ ✤â ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ữỡ tr trữ ữợ Pữỡ tr t❤✉➛♥ ♥❤➜t ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ k + 4k − = ⇒ k1 = 1; k2 = −5 ❉♦ ✤â ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t tữỡ ự ữợ ♣❤÷ì♥❣ tr➻♥❤ ❧➔ f (x) = ex (x + 1)✱ α y = C1 ex + C2 e−5x ❂ ✶ ❧➔ ♠ët ♥❣❤✐➺♠ ✤ì♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ t t ởt r ữỡ tr ữợ ❞↕♥❣✿ Y = xex (Ax + B) = ex (Ax2 + Bx) Y = ex [Ax2 + (B + 2A)x + B] Y = ex [Ax2 + (B + 4A)x + 2A + 2B] ❚❤❛② Y, Y , Y ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ✈➔ rót ❣å♥ t❛ ✤÷đ❝✿ ✶✷❆① ✰✷❆✰✻❇ ❂ ①✰✶ ➜ ⇒    A= 12A = 12 ↔ 2A + 6B =   B= 36 ⑩ ❱➟② ♥❣❤✐➺♠ r✐➯♥❣ ❝➛♥ t➻♠ ❧➔ x Y =e ❿ x2 5x + 12 36 P❍×❒◆● ❚❘➐◆❍ ❱■ P❍❹◆ ✽✷ ⑩ ❉♦ ✤â ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❧➔ ❝✮ trữ ữợ x y = C1 e + C2 e ✳ P❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t t÷ì♥❣ ù♥❣ −5x +e x y − 6y + 9y = ❿ x2 5x + 12 36 ❝â ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ k − 6k + = ⇒ k1 = k2 = ❉♦ ✤â ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr t t tữỡ ự ữợ ✈➳ ♣❤↔✐ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❧➔ f (x) = e3x ✱ α y = C1 e3x + C2 xe3x ❂ ✸ ❧➔ ♠ët ♥❣❤✐➺♠ ❦➨♣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ♥➯♥ t❛ t➻♠ ♠ët ♥❣❤✐➺♠ r✐➯♥❣ ❝õ❛ ữỡ tr ữợ Y = x2 e3x A = Ax2 e3x Y = (2Ax + 3Ax2 )e3x Y = (2A + 12Ax + 9Ax2 )e3x ❚❤❛② Y, Y , Y ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ✈➔ t❛ ✤÷đ❝✿ (2A + 12Ax + 9Ax2 ) − 6(2Ax+3Ax2 ) + 9Ax2 = 1 ⇒ 2A = ⇒ A = ❱➟② ♥❣❤✐➺♠ r✐➯♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❧➔ Y = x2 3x e ❉♦ ✤â ♥❣❤✐➺♠ tờ qt ữỡ tr ã y = C1 e3x + C2 xe3x + x2 3x e ●✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ s❛✉✿ y + y = cos 2x❀ ❜✮ y + y = cos x❀ ❝✮ y + y = cos 2x + cos x ữợ P❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t t÷ì♥❣ ù♥❣ y +y = ❝â ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ k + = ⇒ k = ±i✳ y = C1 cos x + C2 sinx α ± iβ = ± i2 ❦❤ỉ♥❣ ❧➔ ❉♦ ✤â ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t t tữỡ ự ữợ ♣❤÷ì♥❣ tr➻♥❤ ❧➔ 0x f (x) = e cos 2x ❞♦ ✤â ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ✈➔ ❜➟❝ ❝õ❛ ✤❛ t❤ù❝ ❜➡♥❣ ✵ ♥➯♥ t❛ t➻♠ ♠ët ♥❣❤✐➺♠ r ữỡ tr ữợ Y = A cos 2x + B sin 2x Y = −2A sin 2x + 2B cos 2x Y = −4A cos 2x − 4B sin 2x ❚❤❛② Y, Y , Y ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ✈➔ rót ❣å♥ t❛ ✤÷đ❝✿ −3Acos2x − 3Bsin2x = cos2x ➜ −3A = A=− ⇒ ⇒ −3B = B = ❱➟② ♥❣❤✐➺♠ r✐➯♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❧➔ Y = − cos 2x ❉♦ ✤â ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❧➔ y = C1 cos x + C2 sin x − cos 2x ✸✳✷ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ữợ Pữỡ tr t ♥❤➜t t÷ì♥❣ ù♥❣ y +y = ❝â ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ k + = ⇒ k = ±i y¯ = C1 cos x + C2 sinx✳ α ± iβ = ± i ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❉♦ ✤â ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t tữỡ ự ữợ ữỡ tr ❧➔ 0x f (x) = e cos x ❞♦ ✤â ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ✈➔ ❜➟❝ ❝õ❛ ✤❛ t❤ù❝ ❜➡♥❣ ✵ ♥➯♥ t❛ t➻♠ ♠ët ♥❣❤✐➺♠ r✐➯♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ữợ Y = x(A cos x + B sin x) Y = (A + Bx) cos x + (B − Ax) sin x Y = (2B − Ax) cos x − (2A + Bx) sin x ❚❤❛② Y, Y , Y ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ rót ❣å♥ t❛ ✤÷đ❝ ✿ 2Bcosx − (−2Asinx) = cosx ➜ ⇒ 2A = ⇒ 2B = A=0 B= x sin x x ❉♦ ✤â ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❧➔ y = C1 cos x + C2 sin x + sin x ❝✮ ❱➳ ♣❤↔✐ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❧➔ tê♥❣ ❝õ❛ ❤❛✐ ❤➔♠ f1 (x) = cos 2x, f2 (x) = cos x ❱➟② ♥❣❤✐➺♠ r ữỡ tr Y = ỵ ỗ t tờ r ữỡ tr tr ợ ởt r ữỡ tr tr ❧➔ ♠ët ♥❣❤✐➺♠ r✐➯♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦✳ ❱➟② x Y = − cos 2x + sin x ❧➔ ♠ët ♥❣❤✐➺♠ r✐➯♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦✳ ❉♦ ✤â ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❧➔ • ❱➼ ❞ư ✸✳✸✶✳ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ y −y = y = C1 cos x + C2 sin x − x cos 2x + sin x ex + ex ữợ k − = ⇒ k = ±1 y = C1 ex + C2 e−x ✳ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t ❝â ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❱➟② ♥❣❤✐➺♠ tê♥❣ qt ữỡ tr t t tữỡ ự x ữợ ✷ ✳ ❱➳ ♣❤↔✐ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❧➔ f (x) = e + ex ❦❤æ♥❣ ❝â ❞↕♥❣ ✤➦❝ ❜✐➺t✳ ❚❛ sû ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ t❤✐➯♥ ❜➡♥❣ sè ▲❛❣r❛♥❣❡ t ữỡ tr ổ t t ữợ x −x ❞↕♥❣ y = C1 (x)e + C2 (x)e tr♦♥❣ ✤â C1 (x), C2 (x) ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❤➺✿ C1 (x)ex + C2 (x)e−x = C1 (x)ex − C2 (x)e−x = ●✐↔✐ r❛ t❛ ✤÷đ❝ C1 (x) = ex + ex 1 [x − ln(ex + 1) + K1 ]✱ C2 (x) = − [ex − ln(ex + 1) + K2 ] 2 ❱➟② ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❧➔✿ 1 y = ex [x − ln(ex + 1) + K1 ] − e−x [ex − ln(ex + 1) + K2 ] 2 ✽✹ ❇➔✐ t➟♣ ❝❤÷ì♥❣ ✸ ❇➔✐ ✸✳✶✳ ●✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝â ❜✐➳♥ sè ♣❤➙♥ ❧②✿ (1 + x)ydx + (1 − y)xdy = 0❀ (x2 − yx2 )y + y + xy = 0❀ y − y + 1dx√− (1 + x2 )dy = 0❀ ❝✮ 2x ❞✮ x + y dx + y + x2 dy = 0, y|x=0 = 1❀ 2x x ❡✮ (1 + e )y dy = e dx, y|x=0 = 0✳ ❛✮ ❜✮ ✣→♣ sè✿ ln |xy| + x − y = C ❀ y x+y + ln = C❀ ❜✮ xy ⑩x ❿ 2 + y −y+1 ❀ ❝✮ + x = C y − √ √ ❞✮ + x2 + + y = + ❀ 3π x ❡✮ y = 3arctane − ✳ ❛✮ ❇➔✐ ✸✳✷✳ ●✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✤➥♥❣ ❝➜♣ ♠ët✿ 2xy + y ❛✮ y = ❀ x2 ❜✮ xdy − ydx = x2 + y ❀ 2 ❝✮ xyy + x − 2y = 0❀ 2 2 ❞✮ (3x + y )y + (y − x )xy = 0✳ ✣→♣ sè✿ x(x + y) = Cy ❀ + 2Cy√− C x2 = 0❀ ❝✮ y = ±x + C x2 ❀ 2 ❞✮ x(x + y ) − C y = 0✳ ❛✮ ❜✮ ❇➔✐ ✸✳✸✳ ●✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ♠ët✿ 2x − y = 1❀ ❛✮ y − x2 − x −x2 ❜✮ y + 2xy = xe ❀ 2 ❝✮ (1 + x )y − 2xy = (1 + x ) ❀ 2y ❞✮ y − = (x + 1)3 ; y|x=0 = ✳ x+1 ✣→♣ sè✿⑩ ❿ x−1 ❛✮ y = ln + K (x2 − x)❀ x ❿ ⑩ x2 −x2 ❜✮ y = e ❀ C+ 2 ❝✮ y = (1 + x )(x + C)❀ ❿ ⑩ 2 x +x+ ✳ ❞✮ y = (x + 1) 2 ❇➔✐ ✸✳✹✳ ●✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❇❡❝♥✉❧❧② s❛✉✿ √ x(1 + x2 )y − 4x2 y = 2(1 + x2 )2 y ❀ √ ❜✮ y − 2y cot gx = y sin 4x❀ 3 ❝✮ y +xy = x y ❀ x √ ❞✮ y + y = e y , y|x=0 = ✳ ❛✮ ✣→♣ số Pì P Pữỡ tr ❝➜♣ ✷ ✽✺ y = (ln |x| + K)2 (1 + x2 )2 ❀ 2 ❜✮ y = (sin 3x/3 + sin x + K) sin x❀ 2 x ❝✮ y (x + + Ke ) = 1❀ ⑩ ❿2 −x x ❞✮ y = e e +1 ✳ ❛✮ ❇➔✐ ✸✳✺✳ ●✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t♦➔♥ ♣❤➛♥✿ ❛✮ (x + y + 1)dx + (x − y + 3)dy = 0❀ 2 ❜✮ 2(3xy + 2x )dx + 3(2x y + y )dy = 0❀ y y ❝✮ (2x + y)e dx + (x + xy + x)e dy = 0❀ ❿ ⑩ ❞✮ ⑩ ❿ 2y 2x dx + + dy = 0✳ 1+ 2 x +y +1 x + y2 + ✣→♣ sè✿ y3 x2 + x + xy − + 3y = C ❀ 2 ❜✮ x + 3x y + y = C ❀ y ❝✮ (x + xy)e = C ❀ 2 ❞✮ x + y + ln(x + y + 1) = C ✳ ❛✮ ❇➔✐ ✸✳✻✳ ●✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ✷ ❦❤✉②➳t✿ xy − y = x2 ex ❀ y − x(x − 1) = ; y|x=2 = ; y |x=2 = −1❀ ❜✮ y − x−1 ❝✮ y + 2y (1 − 2y) = ; y|x=0 = ; y |x=0 = ❀ ❞✮ xy − y = x ln x ; y|x=1 = − ; y |x=1 = −1✳ ❛✮ ✣→♣ sè✿ y = ex (x − 1) + C1 x2 + C2 ❀ ❜✮ y = (3x4 − 4x3 − 36x2 + 72x + 8)❀ 24 1 ❝✮ y = − ❀ ⑩2(x + 1) ❿ x3 ❞✮ y = ln x − ✳ 3 ❛✮ ❇➔✐ ✸✳✼✳ ●✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ x (ln x − 1)y − xy + y = ❜✐➳t r➡♥❣ ♥â ❝â ♠ët ♥❣❤✐➺♠ r✐➯♥❣ ❞↕♥❣ ②1 ✭①✮ ❂ ①α ✱ α ∈ R✳ αx ❜✮ (2x + 1)y + (4x − 2)y − 8y = ❜✐➳t r➡♥❣ ♥â ❝â ♠ët ♥❣❤✐➺♠ r✐➯♥❣ ❞↕♥❣ ②1 ✭①✮ ❂ ❡ ✱α ∈ R✳ ❝✮ (x − 1)y − 6y = ❜✐➳t r➡♥❣ ♥â ❝â ♠ët ♥❣❤✐➺♠ r✐➯♥❣ ②1 ✭①✮ ❝â ❞↕♥❣ ✤❛ t❤ù❝✳ 2 ❞✮ (2x − x )y + (x − 2)y + 2(1 − x)y = 0, y|x=1 = 0, y |x=1 = ❜✐➳t r➡♥❣ ♥â ❝â ♠ët ♥❣❤✐➺♠ ❛✮ x r✐➯♥❣ ②1 ✭①✮ ❂ ❡ ✳ ❡✮ x ❜✐➳t r➡♥❣ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t t÷ì♥❣ ù♥❣ ❝õ❛ ♥â ❝â −2 ❜✐➳t r➡♥❣ ♥â ❝â ❤❛✐ ♥❣❤✐➺♠ r✐➯♥❣ ❧➔ ②1 ✭①✮ ❂ ✶✱ ②2 ✭①✮ ❂ ①✳ x(x + 1)y + (x + 2)y − y = x + ♠ët ♥❣❤✐➺♠ r✐➯♥❣ ❞↕♥❣ ✤❛ t❤ù❝✳ ❢✮ (2x − x )y + 2(x − 1)y − 2y = ✣→♣ sè✿ y = C1 x + C2 ln |x| ➉❀ ➌ (2x + 1)2 −2x ❜✮ y = C1 e + C2 − 2x ❀ ➉ ➌ 3x(x2 − 1) x+1 ❝✮ y = C1 (x − x) + C2 − x + ln ❀ x−1 x−1 ❞✮ y = x − e ❀ x+2 ❡✮ y = ln |x| + C1 (x + 2) + C2 + ❀ x 2 ❢✮ y = C1 x + C2 (x − 1) + 1✳ ❛✮ Pì P ữỡ tr➻♥❤ √ y + 2y + y = 3e−x x + 1❀ ❜✮ y + y = tgx❀ ❝✮ y + 5y + 6y = ✳ + e2x ✈✐ ♣❤➙♥ s❛✉✿ ❛✮ ✣→♣ sè✿ ⑩ ❿ ❛✮ y = e C1 + C2 x + (1 + x)5/2 ❀ x π ❜✮ y = C1 cos x + C2 sinx − cos x ln tan + ❀ −2x ❝✮ y = C1 e + C2 e−3x + e−2x ln(1 + e2x ) + e−3x arctanex ✳ −x ❇➔✐ ✸✳✾✳ y ❜✮ y ❝✮ y ❞✮ y ❡✮ y ❛✮ ●✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ s❛✉✿ − 7y + 6y = sinx❀ + 9y = 6e3x ❀ − 3y = − 6x❀ − 2y + 3y = e−x cosx❀ + 4y = 2sin2x✳ ✣→♣ sè✿ 5sinx + cos x ❀ 74 3x ❜✮ y = C1 cos3x + C2 sin 3x + e ❀ 3x ❝✮ y = C1 + C2 e + x2 ❀ √ √ e−x x 2x) + (5 cos x − sin x)❀ ❞✮ y = e (C1 cos 2x + C2 sin 41 x ❡✮ y = C1 cos2x + C2 sin 2x − cos2x✳ ❛✮ y = C1 ex + C2 e6x +