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Giải tích toán học tập 3

PHẠM QUANG TRÌNH – NGUYỄN NGỌC ANH
NGUYỄN XUÂN HUY

gI¶I TÝCH TO¸N HäC
TËP 3

NHÀ XUẤT BẢN ĐẠI HỌC QUỐC GIA HÀ NỘI


PHẠM QUANG TRÌNH – NGUYỄN NGỌC ANH
NGUYỄN XUÂN HUY

gI¶I TÝCH TO¸N HäC
TËP 3

NHÀ XUẤT BẢN ĐẠI HỌC QUỐC GIA HÀ NỘI


▼ô❝ ❧ô❝
▼ô❝ ❧ô❝


✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳



✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳



▲ê✐ ♥ã✐ ➤➬✉

❈❤➢➡♥❣ ✶

✶✳✶

✶✳✷

✶✳✸

P❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝✃♣ ✶

❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳



✶✳✶✳✶

P❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝✃♣ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳



✶✳✶✳✷

◆❣❤✐Ö♠



✶✳✶✳✸

❇➭✐ t♦➳♥ ❈❛✉❝❤②


✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵

❙ù tå♥ t➵✐ ✈➭ ❞✉② ♥❤✃t ♥❣❤✐Ö♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵

✶✳✷✳✶

➜✐Ò✉ ❦✐Ö♥ ▲✐♣s❝❤✐t③ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵

✶✳✷✳✷

❉➲② ①✃♣ ①Ø P✐❝❛r

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶

✶✳✷✳✸

➜Þ♥❤ ❧ý tå♥ t➵✐ ✈➭ ❞✉② ♥❤✃t ♥❣❤✐Ö♠ ✭❈❛✉❝❤②✲P✐❝❛r✮ ✳ ✳ ✳

✶✷

✶✳✷✳✹

❙ù t❤➳❝ tr✐Ó♥ ♥❣❤✐Ö♠

✶✻

✶✳✷✳✺

❈➳❝ ❧♦➵✐ ♥❣❤✐Ö♠ ❝ñ❛ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝✃♣ ✶

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳

✶✻

P❤➢➡♥❣ ♣❤➳♣ ❣✐➯✐ ♠ét sè ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝✃♣ ✶ ✳ ✳ ✳ ✳ ✳ ✳

✶✼

✶✳✸✳✶

P❤➢➡♥❣ tr×♥❤ ♣❤➞♥ ❧✐ ❜✐Õ♥ sè

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✼

✶✳✸✳✷

P❤➢➡♥❣ tr×♥❤ t❤✉➬♥ ♥❤✃t

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✾

✶✳✸✳✸

P❤➢➡♥❣ tr×♥❤ q✉② ➤➢î❝ ✈Ò ♣❤➢➡♥❣ tr×♥❤ t❤✉➬♥ ♥❤✃t

✳ ✳ ✳

✷✶

✶✳✸✳✹

P❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ t♦➭♥ ♣❤➬♥✳ ❚❤õ❛ sè tÝ❝❤ ♣❤➞♥

✳ ✳ ✳

✷✹

✶✳✸✳✺

P❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ t✉②Õ♥ tÝ♥❤✱ ♣❤➢➡♥❣ tr×♥❤ ❇❡r♥♦✉❧❧✐
✈➭ ♣❤➢➡♥❣ tr×♥❤ ❘✐❝❛t✐

✶✳✹



✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✾

❇➭✐ t❐♣ ❝❤➢➡♥❣ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✻






▼Ô❈ ▲Ô❈

❈❤➢➡♥❣ ✷

✷✳✶

✷✳✷

P❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝✃♣ ❝❛♦

❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥

✸✾

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✾

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✵

✷✳✶✳✶

◆❣❤✐Ö♠

✷✳✶✳✷

❇➭✐ t♦➳♥ ❈❛✉❝❤②

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✵

❙ù tå♥ t➵✐ ✈➭ ❞✉② ♥❤✃t ♥❣❤✐Ö♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✵

✷✳✷✳✶

➜✐Ò✉ ❦✐Ö♥ ▲✐♣s❝❤✐t③ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✶

✷✳✷✳✷

➜Þ♥❤ ❧ý tå♥ t➵✐ ✈➭ ❞✉② ♥❤✃t ♥❣❤✐Ö♠

✹✶

✷✳✷✳✸

❈➳❝ ❧♦➵✐ ♥❣❤✐Ö♠ ❝ñ❛ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝✃♣ ♥

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳

✹✷

✷✳✸

P❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ t✉②Õ♥ tÝ♥❤ ❝✃♣ ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✸

✷✳✹

P❤➢➡♥❣ tr×♥❤ t✉②Õ♥ tÝ♥❤ t❤✉➬♥ ♥❤✃t ❝✃♣ ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✹

✷✳✹✳✶

▼ét sè tÝ♥❤ ❝❤✃t ❝ñ❛ ♥❣❤✐Ö♠ ♣❤➢➡♥❣ tr×♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✺

✷✳✹✳✷

❙ù ♣❤ô t❤✉é❝ t✉②Õ♥ tÝ♥❤ ✈➭ ➤é❝ ❧❐♣ t✉②Õ♥ tÝ♥❤ ❝ñ❛ ❤Ö ❤➭♠ ✹✺

✷✳✹✳✸

➜Þ♥❤ t❤ø❝ ❱r♦♥s❦✐

✷✳✹✳✹

❈➠♥❣ t❤ø❝ ❖str♦❣r❛❞s❦✐ ✲ ▲✐✉✈✐❧

✷✳✹✳✺

❍Ö ♥❣❤✐Ö♠ ❝➡ ❜➯♥✱ ♥❣❤✐Ö♠ tæ♥❣ q✉➳t

✷✳✺

✷✳✻

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✼
✺✵
✺✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✻

✷✳✺✳✶

◆❣❤✐Ö♠

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✻

✷✳✺✳✷

P❤➢➡♥❣ ♣❤➳♣ ❜✐Õ♥ t❤✐➟♥ ❤➺♥❣ sè ✭▲❛❣r❛♥❣❡✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✽

P❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ t✉②Õ♥ tÝ♥❤ ❝✃♣ ✷ ❤Ö sè ❤➺♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✾

◆❣❤✐Ö♠ ❝ñ❛ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ t✉②Õ♥ tÝ♥❤ t❤✉➬♥ ♥❤✃t
❝✃♣ ❤❛✐ ❤Ö sè ❤➺♥❣

✷✳✻✳✷

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻✵

◆❣❤✐Ö♠ ❝ñ❛ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ t✉②Õ♥ tÝ♥❤ ❦❤➠♥❣ t❤✉➬♥
♥❤✃t ❝✃♣ ❤❛✐ ❤Ö sè ❤➺♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻✷

❇➭✐ t❐♣ ❝❤➢➡♥❣ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻✾

❈❤➢➡♥❣ ✸

❍Ö ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥

✸✳✶

❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥

✸✳✷

❇➭✐ t♦➳♥ ❈❛✉❝❤②
✸✳✷✳✶

✸✳✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

P❤➢➡♥❣ tr×♥❤ t✉②Õ♥ tÝ♥❤ ❦❤➠♥❣ t❤✉➬♥ ♥❤✃t ❝✃♣ ♥

✷✳✻✳✶

✷✳✼

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼✶

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼✶

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼✷

❇➭✐ t♦➳♥ ❈❛✉❝❤②

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼✷

P❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝✃♣ ❝❛♦ ✈➭ ❤Ö ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝✃♣ ♠ét ✼✷
✸✳✸✳✶

➜➢❛ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝✃♣ ♥ ✈Ò ❤Ö ♥ ♣❤➢➡♥❣ tr×♥❤ ✈✐
♣❤➞♥ ❝✃♣ ♠ét

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼✷


▼ô❝ ❧ô❝



✸✳✸✳✷

➜➢❛ ❤Ö ♥ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝✃♣ ♠ét ✈Ò ♠ét ♣❤➢➡♥❣
tr×♥❤ ✈✐ ♣❤➞♥ ❝✃♣ ♥

✸✳✹

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✳✸✳✸

➜Þ♥❤ ❧ý tå♥ t➵✐ ✈➭ ❞✉② ♥❤✃t ♥❣❤✐Ö♠

✸✳✸✳✹

❙ù t❤➳❝ tr✐Ó♥ ♥❣❤✐Ö♠

✸✳✸✳✺

❈➳❝ ❧♦➵✐ ♥❣❤✐Ö♠ ❝ñ❛ ❤Ö ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽✵

❍Ö ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ t✉②Õ♥ tÝ♥❤

✳ ✳ ✳ ✳ ✳ ✳

✽✵

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽✷

✸✳✹✳✶

❍Ö ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ t✉②Õ♥ tÝ♥❤ t❤✉➬♥ ♥❤✃t

✸✳✹✳✷

❈➳❝ tÝ♥❤ ❝❤✃t ❝ñ❛ ❤Ö ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ t✉②Õ♥ tÝ♥❤

✳ ✳ ✳ ✳ ✳

t❤✉➬♥ ♥❤✃t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸✳✹✳✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽✸

✽✸

✸✳✹✳✹

❍Ö ♥❣❤✐Ö♠ ❝➡ ❜➯♥

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽✻

✸✳✹✳✺

❍Ö ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ t✉②Õ♥ tÝ♥❤ ❦❤➠♥❣ t❤✉➬♥ ♥❤✃t ✳ ✳

✽✽

✸✳✹✳✻

❈➳❝ tÝ♥❤ ❝❤✃t ♥❣❤✐Ö♠ ❝ñ❛ ❤Ö ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ t✉②Õ♥
tÝ♥❤ ❦❤➠♥❣ t❤✉➬♥ ♥❤✃t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽✾

✸✳✹✳✼

◆❣❤✐Ö♠ tæ♥❣ q✉➳t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽✾

✸✳✹✳✽

P❤➢➡♥❣ ♣❤➳♣ ❜✐Õ♥ t❤✐➟♥ ❤➺♥❣ sè ✭▲❛❣r❛♥❣❡✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾✵

❍Ö ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ t✉②Õ♥ tÝ♥❤ ❤Ö sè ❤➺♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾✸

✸✳✺✳✶

❍Ö ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ t✉②Õ♥ tÝ♥❤ t❤✉➬♥ ♥❤✃t ❤Ö sè ❤➺♥❣ ✾✸

✸✳✺✳✷

◆❣❤✐Ö♠ ❝ñ❛ ❤Ö ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ t✉②Õ♥ tÝ♥❤ t❤✉➬♥
♥❤✃t ❤Ö sè ❤➺♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✳✺✳✸

✾✹

❍Ö ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ t✉②Õ♥ tÝ♥❤ ❦❤➠♥❣ t❤✉➬♥ ♥❤✃t ❤Ö
sè ❤➺♥❣

✸✳✻

✽✷

❙ù ♣❤ô t❤✉é❝ t✉②Õ♥ ✈➭ ➤é❝ ❧❐♣ t✉②Õ♥ tÝ♥❤ ❝ñ❛ ❤Ö ✈Ð❝t➡
❤➭♠

✸✳✺

✼✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾✾

❇➭✐ t❐♣ ❝❤➢➡♥❣ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✷




●✐➯✐ tÝ❝❤ ❚♦➳♥ ❤ä❝


ờ ó
ộ trì tí ọ ồ t ợ s ở t
tể t P rì s ễ s ễ
ọ ự t trì tí ọ ợ ộ
ồ ộ ủ ộ ụ t t ị ù trờ
ọ ứ t ợ ệ q t s
trờ ệ ĩ tt ọ
ộ trì ợ s t ị ớ ọ ọ
ù ợ ớ tờ t ứ ọ ù ợ ớ ố
tợ s ệ ĩ tt t ột rõ ét ệ
ụ ết q ý tết ồ tờ ột tốt t tí
ọ ủ ệ tố ế tứ tr trì
ủ ộ trì ệ tố ế tứ ề
trì ệ trì ợ ớ tệ tr
P trì
P trì
ệ trì
ọ trì ũ t ệ t tốt s

t rt ố ợ sự ó ý qý ủ ồ
ệ ọ ể ộ s ợ tệ t
ớ tệ ộ s tớ ọ




●✐➯✐ tÝ❝❤ t♦➳♥ ❤ä❝


❈❤➢➡♥❣ ✶
P❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝✃♣ ✶
✶✳✶

❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥

✶✳✶✳✶

P❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝✃♣ ✶

P❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝✃♣ ✶ ❞➵♥❣ tæ♥❣ q✉➳t

F (x, y, y ) = 0

✭✶✳✶✮

F ①➳❝ ➤Þ♥❤ tr♦♥❣ ♠✐Ò♥ G ⊂ R3 ✳
◆Õ✉ tr♦♥❣ ♠✐Ò♥ G✱ tõ ♣❤➢➡♥❣ tr×♥❤ ✭✶✳✶✮ t❛ ❝ã t❤Ó ❣✐➯✐ ➤➢î❝ y
❚r♦♥❣ ➤ã ❤➭♠

y = f (x, y)

✭✶✳✷✮

t❤× t❛ ➤➢î❝ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝✃♣ ✶ ➤➲ ❣✐➯✐ r❛ ➤➵♦ ❤➭♠✳

❱Ý ❞ô✳

✶✳✶✳✷

yy = x2 + y 2 ,

y = xy + y 2 ,

dy
= 2y ✳
dx

◆❣❤✐Ö♠

❍➭♠ sè

y = ϕ(x)

①➳❝ ➤Þ♥❤ ✈➭ ❦❤➯ ✈✐ tr➟♥ ❦❤♦➯♥❣

❣ä✐ ❧➭ ♥❣❤✐Ö♠ tæ♥❣ q✉➳t ❝ñ❛ ♣❤➢➡♥❣ tr×♥❤ ✭✶✳✶✮ ♥Õ✉
❛✮

(x, ϕ(x), ϕ (x)) ∈ G

✈í✐ ♠ä✐

x ∈ I✳

I = (a, b)

➤➢î❝


✶✵

P❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝✃♣ ✶

❜✮

F (x, ϕ(x), ϕ (x)) ≡ 0

tr➟♥ I ✳
dy
❱Ý ❞ô✳ ❳Ðt ♣❤➢➡♥❣ tr×♥❤
= 2y
dx
①➳❝ ➤Þ♥❤ tr➟♥ ❦❤♦➯♥❣ (−∞, +∞)

❝ã t❤Ó ❦✐Ó♠ tr❛ trù❝ t✐Õ♣
✈í✐

y = ce2x

❧➭ ❤➺♥❣ sè t✉ú ý ❧➭ ♥❣❤✐Ö♠

c

❝ñ❛ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ➤➲ ❝❤♦✳

✶✳✶✳✸

❇➭✐ t♦➳♥ ❈❛✉❝❤②

◗✉❛ ✈Ý ❞ô tr➟♥ t❛ t❤✃② ◆❣❤✐Ö♠ ❝ñ❛ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❧➭ ✈➠
sè ✭❞♦ ❤➺♥❣ sè

c

❝ã t❤Ó ❧✃② t✉ú ý✮✳ ❚r♦♥❣ t❤ù❝ tÕ t❛ t❤➢ê♥❣ q✉❛♥

t➞♠ ➤Õ♥ ♥❣❤✐Ö♠ ❝ñ❛ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ t❤♦➯ ♠➲♥ ♥❤÷♥❣ ➤✐Ò✉
❦✐Ö♥ ♥➭♦ ➤ã✱ ❝❤➻♥❣ ❤➵♥

y(x0 ) = y0 .

✭✶✳✸✮

➜✐Ò✉ ❦✐Ö♥ tr➟♥ ➤➢î❝ ❣ä✐ ❧➭ ➤✐Ò✉ ❦✐Ö♥ ❜❛♥ ➤➬✉✳ ❇➭✐ t♦➳♥ t×♠ ♥❣❤✐Ö♠
❝ñ❛ ♣❤➢➡♥❣ tr×♥❤ ✭✶✳✶✮ ❤♦➷❝ ✭✶✳✷✮ t❤♦➯ ♠➲♥ ➤✐Ò✉ ❦✐Ö♥ ❜❛♥ ➤➬✉ ✭✶✳✸✮
❣ä✐ ❧➭ ❜➭✐ t♦➳♥ ❈❛✉❝❤②✳ ❚❛ sÏ t×♠ ❝➳❝ ➤✐Ò✉ ❦✐Ö♥ ➤Ó ❜➭✐ t♦➳♥ ❈❛✉❝❤②
❝ã ♥❣❤✐Ö♠ ❞✉② ♥❤✃t✳

✶✳✷

❙ù tå♥ t➵✐ ✈➭ ❞✉② ♥❤✃t ♥❣❤✐Ö♠

❳Ðt ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥

y = f (x, y),
tr♦♥❣ ➤ã
❝ñ❛

f

f

①➳❝ ➤Þ♥❤ tr♦♥❣ ♠✐Ò♥

G ⊂ R2 ✳

✭✶✳✹✮

❚❛ sÏ ❝❤Ø r❛ ❝➳❝ ➤✐Ò✉ ❦✐Ö♥

➤Ó ❜➭✐ t♦➳♥ ❈❛✉❝❤② ø♥❣ ✈í✐ ♣❤➢➡♥❣ tr×♥❤ ✭✶✳✹✮ ❝ã ♥❣❤✐Ö♠

❞✉② ♥❤✃t✳

✶✳✷✳✶

➜✐Ò✉ ❦✐Ö♥ ▲✐♣s❝❤✐t③

❍➭♠

f

①➳❝ ➤Þ♥❤ tr♦♥❣ ♠✐Ò♥

❝❤✐t③ t❤❡♦ ❜✐Õ♥

y

G

❣ä✐ ❧➭ t❤♦➯ ♠➲♥ ➤✐Ò✉ ❦✐Ö♥ ▲✐♣s✲

♥Õ✉ tå♥ t➵✐ ❤➺♥❣ sè

L>0

s❛♦ ❝❤♦ ✈í✐ ❤❛✐ ➤✐Ó♠


ự tồ t t ệ

(x, y), (x, y ) G



t ỳ t ó t tứ

|f (x, y) f (x, y )| L|y y |.
ú ý ế

G

f

ó r t

y



ị tr ề

tì t ề ệ st ộ tự ứ ù

ị í r



ỉ Pr

sử


G

f (x, y)

tụ tr ề

ọ số

a, b

G (x0 , y0 )

ể tr

s ì ữ t

Q = {|x x0 | a, |y y0 | b}
ứ tr

G

t

M = max{|f (x, y)| : (x, y) Q} h = min{a,

b
}.
M

ự ệ ỉ ủ trì
s

y0 (x) = y0
x

y1 (x) = y0 +

f (, y0 ( )d,

x [x0 h, x0 + h]

x0

ããã
x

yn (x) = y0 +

f (, yn1 ( )d,

x [x0 h, x0 + h].

x0



yn (x)

ị tr ợ ọ ỉ Pr

[x0 h, x0 + h] tì (x, yn (x)) Q,
n = 0, 1, 2, ã ã ã
t ề ú ớ n = 0
sử t ó (x, yn1 (x)) Q x [x0 h, x0 + h] ó t ó


x

ế t tr

tể ự

x

yn (x) = y0 +

f (, yn1 ( )d.
x0




P trì



|x x0 | h a

t ó
x

f (, yn1 ( )d |

|yn (x) y0 | = |
x0
x

|f (, yn1 ( )|d |

|
x0

x

d | = M |x x0 |

M|
x0

Mh M
tứ

(x, yn (x)) Q





b
=b
M

x [x0 h, x0 + h]

ị ý tồ t t ệ Pr

ị í

sử

f

t ề ệ s



f

tụ tr ề

G



f

t ề ệ st t ế

y

tr

G

(x0 , y0 ) G tồ t t ột ệ y = y(x)
ủ trì t ề ệ y(x0 ) = y0 ệ
ị tr ột ó [x0 h, x0 + h] ủ x0 tr ó h
số ị ụ tộ f ể (x0 , y0 ) ề G
ó ứ ớ ỗ ể

ứ ét ỉ Pr

{yn (x)}

ự ở tr

(x, yn (x)) Q, n f tụ yn (x) tụ
tr [x0 h, x0 + h] ễ t yn (x0 ) = y0 , n. ờ t ứ
yn (x) ộ tụ ề tr [x0 h, x0 + h] r [x0 h, x0 + h]

ì


✶✳✷ ❙ù tå♥ t➵✐ ✈➭ ❞✉② ♥❤✃t ♥❣❤✐Ö♠

✶✸

t❛ ❝ã
x

f (τ, y0 )dτ ≤ M |x − x0 |,

|y1 (x) − y0 (x)| =
x0
x

[f (τ, y1 (τ )) − f (τ, y0 (τ ))]dτ

|y2 (x) − y1 (x)| =
x0
x

|f (τ, y1 (τ )) − f (τ, y0 (τ ))|dτ


x0

x

|y1 (τ ) − y0 (τ )|dτ

≤ L
x0

x

≤ ML

|τ − x0 |dτ =

ML
|x − x0 |2 .
2!

x0

x ∈ [x0 − h, x0 + h]

❚❛ sÏ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ❦❤✐

|yn (x) − yn−1 (x)| ≤
❚❤❐t ✈❐②✱ ✈í✐
✈í✐

n✳

n = 1, 2, · · · ✱

t❤×

M Ln−1
|x − x0 |n .
n!

✭✶✳✻✮

t❛ ➤➲ ❦✐Ó♠ tr❛ ë tr➟♥✳ ●✐➯ sö ✭✶✳✻✮ ➤ó♥❣

❑❤✐ ➤ã
x

|yn+1 (x) − yn (x)| =

[f (τ, yn (τ )) − f (τ, yn−1 (τ ))]dτ
x0
x

≤ L

|yn (τ ) − yn−1 (τ )|dτ
x0

M Ln

n!

x

|τ − x0 |n dτ =

M Ln
|x − x0 |n+1
n!

x0

tø❝ ❧➭ ❜✃t ➤➻♥❣ t❤ø❝ ✭✶✳✻✮ ➤ó♥❣ ✈í✐
❚õ ➤ã✱ ✈í✐

n + 1✳
∀x ∈ [x0 − h, x0 + h], ∀n = 1, 2, · · ·
|yn (x) − yn−1 (x)| ≤

t❛ ❝ã

M Ln−1 n
h .
n!

✭✶✳✼✮


✶✹

P❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝✃♣ ✶

❳Ðt ❝❤✉ç✐ ❤➭♠

y0 (x) + y1 (x) − y0 (x) + · · · + yn (x) − yn−1 (x) + · · ·

✭✶✳✽✮

❉♦ ✭✶✳✼✮ ❣✐➳ trÞ t✉②Öt ➤è✐ ❝ñ❛ sè ❤➵♥❣ tæ♥❣ q✉➳t ❝❤✉ç✐ tr➟♥
❦❤➠♥❣ ✈➢ît q✉➳ sè ❤➵♥❣ tæ♥❣ q✉➳t ❝ñ❛ ❝❤✉ç✐ sè ❞➢➡♥❣ ❤é✐ tô

M Ln−1 n
h . ❚❤❡♦ t✐➟✉ ❝❤✉➮♥ ❲❡✐❡rstr❛ss✱ ❝❤✉ç✐ ✭✶✳✽✮ ❤é✐ tô ➤Ò✉
n!
n=1
tr➟♥ [x0 − h, x0 + h] ➤Õ♥ ♠ét ❤➭♠ y(x) ♥➭♦ ➤ã✳ ❉Ô t❤✃② r➺♥❣ tæ♥❣
r✐➟♥❣ t❤ø n ❝ñ❛ ❝❤✉ç✐ ✭✶✳✽✮ ❧➭ yn (x) ♥➟♥ t❛ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ➤➢î❝


tr➟♥

yn (x) ⇒ y(x)

[x0 − h, x0 + h].

❱×
x

yn (x) = y0 +

f (τ, yn−1 (τ )dτ

✭✶✳✾✮

x0

✈➭

f

❧➭ ❤➭♠ ❧✐➟♥ tô❝ tr➟♥

G

♥➟♥ ❝❤✉②Ó♥ q✉❛ ❣✐í✐ ❤➵♥ ❦❤✐

n→∞

❞➢í✐ ❞✃✉ tÝ❝❤ ♣❤➞♥ ➤➻♥❣ t❤ø❝ tr➟♥ t❛ ❝ã
x

y(x) = y0 +

f (τ, y(τ )dτ.

✭✶✳✶✵✮

x0

❉♦ sù ❤é✐ tô ❝ñ❛ ❞➲②

{yn (x)}

❧➭ ➤Ò✉ tr➟♥ ➤♦➵♥

[x0 − h, x0 + h]

♥➟♥

[x0 − h, x0 + h]✳ ➜➻♥❣ t❤ø❝ ✭✶✳✶✵✮ ✈➭
sù ❧✐➟♥ tô❝ ❝ñ❛ ❤➭♠ f ❝❤♦ t❛ tÝ♥❤ ❦❤➯ ✈✐ ❝ñ❛ y(x) tr➟♥ [x0 − h, x0 + h]✳
❤➭♠ ❣✐í✐ ❤➵♥

y(x)

❧✐➟♥ tô❝ tr➟♥

▲✃② ➤➵♦ ❤➭♠ ❤❛✐ ✈Õ ❝ñ❛ ✭✶✳✶✵✮ t❛ ❝ã

∀x ∈ [x0 − h, x0 + h].

y (x) = f (x, y(x)),
❍✐Ó♥ ♥❤✐➟♥
➤Þ♥❤ tr➟♥

y(x0 ) = y0 ♥➟♥ y(x)
[x0 − h, x0 + h]✳

❧➭ ♥❣❤✐Ö♠ ❝ñ❛ ❜➭✐ t♦➳♥ ❈❛✉❝❤② ①➳❝

❇➞② ❣✐ê t❛ ❝❤ø♥❣ ♠✐♥❤ ♥❣❤✐Ö♠ ♥➭② ❧➭ ❞✉② ♥❤✃t✳ ●✐➯ sö ♣❤➢➡♥❣
tr×♥❤ ✭✶✳✹✮ ❝ß♥ ❝ã ♥❣❤✐Ö♠

y(x)

✈➭ t❤á❛ ♠➲♥ ➤✐Ò✉ ❦✐Ö♥ ❜❛♥ ➤➬✉

y (x) = f (x, y(x),

①➳❝ ➤Þ♥❤ tr➟♥ ➤♦➵♥

y(x0 ) = y0 ✳

[x0 − h , x0 + h ]

❑❤✐ ➤ã

∀x ∈ [x0 − h , x0 + h ].


✶✳✷ ❙ù tå♥ t➵✐ ✈➭ ❞✉② ♥❤✃t ♥❣❤✐Ö♠

✶✺

❚Ý❝❤ ♣❤➞♥ ➤➻♥❣ t❤ø❝ ♥➭② tr➟♥ ➤♦➵♥

[x0 , x]

✈í✐

x ∈ [x0 − h , x0 + h ]

t❛

❝ã
x

f (τ, y n−1 (τ ))dτ.

y n (x) = y0 +

✭✶✳✶✶✮

x0

δ = min{h, h }
[x0 − δ, x0 + δ] t❛ ❝ã
➜➷t

✈➭ ①Ðt ➤➻♥❣ t❤ø❝ ✭✶✳✾✮✱ ✭✶✳✶✶✮ tr➟♥ ➤♦➵♥

x

y(x) − yn (x) =

[f (τ, y(τ )) − f (τ, yn−1 (τ ))]dτ.
x0

❚❛ sÏ ❝❤ø♥❣ ♠✐♥❤

|y(x) − yn (x)| ≤
❚❤❐t ✈❐②✱ ✈í✐

n=0

M Ln n+1
δ ,
(n + 1)!

∀n.

t❛ ❝ã
x

|y(x) − y0 (x)| = |y(x) − y0 | =

[f (τ, y(τ ))
x0

≤ M |x − x0 | ≤ M δ.
●✐➯ sö ✭✶✳✶✷✮ ➤ó♥❣ ✈í✐

n

tø❝ ❧➭

|y(x) − yn (x)| ≤
❚❛ ❝❤ø♥❣ ♠✐♥❤ ♥ã ➤ó♥❣ ✈í✐

M Ln n+1
δ .
(n + 1)!

(n + 1)✳

❉Ô t❤✃②

x

|yn+1 (x) − y(x)| =

[f (τ, yn (τ )) − f (τ, y(τ ))]dτ
x0

M Ln+1

(n + 1)!

x

|τ − x0 |n+1 dτ
x0

n+1

ML
|x − x0 |n+2
(n + 1)!
M Ln+1 n+2

δ .
(n + 2)!
=

✭✶✳✶✷✮




P trì

ợ ứ

M Ln n+1

=0
n (n + 1)!
lim



lim yn (x) = y(x),

n

x [x0 , x0 + ]

tí t ủ ớ

y(x) y(x)

tr

[x0 , x0 + ]

ị í ợ ứ t



ự t trể ệ

ị ý tr t ệ

y = y(x) ủ trì
ớ ề ệ y(x0 ) = y0 tr ột ủ ể
x0 ĩ t {(x, y(x))|x [x0 h, x0 + h]} ột t ủ
G ờ t ứ ợ ó tể t trể é ệ
y = y(x) ó s t {(x, y(x))} ó ớ é tỳ ý ủ
ề G ó t ớ trì ợ
ỉ tr ề G


ệ ủ trì

ét trì

y = f (x, y)

ệ tổ qt ệ tổ qt ủ trì

y = (x, C)

t

ừ ệ tứ

y0 = (x0 , C)

ó tể r

C = (x0 , y0 )

ớ ỗ

(x0 , y0 ) G
ệ tứ

y = (x, C) ệ ủ trì y = f (x, y)
ớ ỗ trị ủ C ợ ị tr
í ụ ễ ể tr ợ trì y = y ó ệ tổ
qt y = Cex
í tổ qt ề trì t
ế ệ tứ

(x, y, C) = 0

ệ tứ ợ ọ tí


✶✳✸ P❤➢➡♥❣ ♣❤➳♣ ❣✐➯✐ ♠ét sè ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝✃♣ ✶

tæ♥❣ q✉➳t ❝ñ❛ ♣❤➢➡♥❣ tr×♥❤ ➤➲ ❝❤♦ tr♦♥❣ ♠✐Ò♥

G

✶✼

♥Õ✉ tr♦♥❣ ➤ã ①➳❝

➤Þ♥❤ ♥❣❤✐Ö♠ tæ♥❣ q✉➳t
❱Ý ❞ô✳ P❤➢➡♥❣ tr×♥❤

y = ϕ(x, C) ❝ñ❛ ♣❤➢➡♥❣ tr×♥❤ ❜❛♥ ➤➬✉✳
y = −x/y, (y = 0) ❝ã tÝ❝❤ ♣❤➞♥ tæ♥❣ q✉➳t ❧➭
x2 + y 2 = C, C > 0

✈× tr♦♥❣ ♥ö❛ ♠➷t ♣❤➻♥❣ ♣❤Ý❛ tr➟♥ ♥ã ①➳❝ ➤Þ♥❤ ♥❣❤✐Ö♠ tæ♥❣ q✉➳t


C − x2 ✱ tr♦♥❣ ♥ö❛ ♠➷t

tæ♥❣ q✉➳t y = − C − x2 ✳
y=

♣❤➻♥❣ ♣❤Ý❛ ❞➢í✐ ♥ã ①➳❝ ➤Þ♥❤ ♥❣❤✐Ö♠

❝✳ ◆❣❤✐Ö♠ r✐➟♥❣✳ ◆❣❤✐Ö♠ r✐➟♥❣ ❝ñ❛ ♣❤➢➡♥❣ tr×♥❤ ❧➭ ♥❣❤✐Ö♠ ♠➭
t➵✐ ♠ç✐ ➤✐Ó♠✱ tÝ♥❤ ❞✉② ♥❤✃t ♥❣❤✐Ö♠ ❝ñ❛ ❜➭✐ t♦➳♥ ❈❛✉❝❤② ➤➢î❝ ❜➯♦
➤➯♠✳ ◆❣❤✐Ö♠ r✐➟♥❣ ♥❤❐♥ ➤➢î❝ tõ ♥❣❤✐Ö♠ tæ♥❣ q✉➳t ❜➺♥❣ ❝➳❝❤ ①➳❝
➤Þ♥❤ ❤➺♥❣ sè ❈ t❤❡♦ ❝➳❝ ➤✐Ò✉ ❦✐Ö♥ ❜❛♥ ➤➬✉✳
❞✳ ◆❣❤✐Ö♠ ❦× ❞Þ✳ ◆❣❤✐Ö♠ ❦× ❞Þ ❧➭ ♥❣❤✐Ö♠ ♠➭ t➵✐ ♠ç✐ ➤✐Ó♠ ❝ñ❛ ♥ã✱
tÝ♥❤ ❞✉② ♥❤✃t ♥❣❤✐Ö♠ ❝ñ❛ ❜➭✐ t♦➳♥ ❈❛✉❝❤② ❜Þ ♣❤➳ ✈ì✳

✶✳✸

P❤➢➡♥❣

♣❤➳♣

❣✐➯✐

♠ét



♣❤➢➡♥❣

tr×♥❤

✈✐

♣❤➞♥ ❝✃♣ ✶

✶✳✸✳✶

P❤➢➡♥❣ tr×♥❤ ♣❤➞♥ ❧✐ ❜✐Õ♥ sè

❛✳ ❉➵♥❣ tæ♥❣ q✉➳t

f (x)dx = g(y)dy.

✭✶✳✶✸✮

P❤➢➡♥❣ ♣❤➳♣ ❣✐➯✐✳ ▲✃② tÝ❝❤ ♣❤➞♥ ✷ ✈Õ t❛ ➤➢î❝

f (x)dx =

g(y)dy.

➜➻♥❣ t❤ø❝ ♥➭② ❝❤♦ t❛ tÝ❝❤ ♣❤➞♥ tæ♥❣ q✉➳t ❝ñ❛ ♣❤➢➡♥❣ tr×♥❤ ✭✶✳✶✸✮✳
❱í✐ ➤✐Ò✉ ❦✐Ö♥ ❜❛♥ ➤➬✉

y0 = y(x0 )✱

tÝ❝❤ ♣❤➞♥ tæ♥❣ q✉➳t ➤➢î❝ ✈✐Õt

x

y

❞➢í✐ ❞➵♥❣

f (τ )dτ =
x0

g(η)dη.
y0


✶✽

P❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝✃♣ ✶

❱Ý ❞ô✳ P❤➢➡♥❣ tr×♥❤

2x
2y
dx +
dy = 0
2
1+x
1 + y2

❝ã tÝ❝❤ ♣❤➞♥ tæ♥❣ q✉➳t

❧➭

2x
dx +
1 + x2

2y
dy = C
1 + y2

❤❛②

ln(1 + x2 ) + ln(1 + y 2 ) = C, C > 0.
❚❛ ❝ò♥❣ ❝ã t❤Ó ✈✐Õt
❝❤♦ tr➢í❝ t❤× ❤➺♥❣

(1 + x2 )(1 + y 2 ) = C , C = eC ✳
sè C, C tr♦♥❣ ❝➳❝ ❝➠♥❣ t❤ø❝

❱í✐ ➤✐Ò✉ ❦✐Ö♥ ➤➬✉
tr➟♥ ①➳❝ ➤Þ♥❤✳

❈❤ó ý✳ ❙❛✉ ♥➭② ❦❤✐ ❣✐➯✐ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝✃♣ ✶ t❤ù❝ ❝❤✃t ❧➭
t×♠ ❝➳❝❤ ➤➢❛ ♣❤➢➡♥❣ tr×♥❤ ➤❛♥❣ ①Ðt ✈Ò ❞➵♥❣ ♣❤➢➡♥❣ tr×♥❤ ♣❤➞♥ ❧✐
❜✐Õ♥ sè✳ ▼ét ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝✃♣ ✶ ①❡♠ ♥❤➢ ➤➲ ❣✐➯✐ ①♦♥❣
♥Õ✉ t❛ ♣❤➞♥ ❧✐ ➤➢î❝ ❜✐Õ♥ sè✳
❜✳ P❤➢➡♥❣ tr×♥❤ ✈í✐ ❜✐Õ♥ sè ♣❤➞♥ ❧✐ ➤➢î❝✳ ➜ã ❧➭ ♣❤➢➡♥❣ tr×♥❤ ❞➵♥❣

m1 (x)n1 (y)dx = m2 (x)n2 (y)dy.
●✐➯ sö

n1 (y)m2 (x) = 0✳

✭✶✳✶✹✮

❈❤✐❛ ❤❛✐ ✈Õ ❝❤♦ ❜✐Ó✉ t❤ø❝ ♥➭② t❛ ➤➢î❝

♣❤➢➡♥❣ tr×♥❤ ♣❤➞♥ ❧✐ ❜✐Õ♥ sè✱ ❜➭✐ t♦➳♥ ➤➢î❝ ❣✐➯✐ ①♦♥❣✳ ❈➳❝ ❣✐➳ trÞ
❝ñ❛

x, y

❧➭♠ ❝❤♦

n1 (y)m2 (x) = 0

❝ò♥❣ ❧➭ ♥❣❤✐Ö♠ ❝ñ❛ ♣❤➢➡♥❣ tr×♥❤

✭✶✳✶✹✮✳
❱Ý ❞ô✳ ❳Ðt ♣❤➢➡♥❣ tr×♥❤


x 1 − y 2 dx + y 1 − x2 dy = 0.
●✐➯ sö


1 − x2

1 − y 2 = 0✳

❈❤✐❛ ✷ ✈Õ ❝ñ❛ ♣❤➢➡♥❣ tr×♥❤ ❝❤♦ ❜✐Ó✉

t❤ø❝ ♥➭② t❛ ➤➢î❝ ♣❤➢➡♥❣ tr×♥❤ ♣❤➞♥ ❧✐ ❜✐Õ♥ sè ✈➭ ❝ã t❤Ó ①➳❝ ➤Þ♥❤
➤➢î❝ tÝ❝❤ ♣❤➞♥ tæ♥❣ q✉➳t ❧➭

1 − y2 +



1 − x2 = C,

C > 0.

1 − x2 . 1 − y 2 = 0 ❝❤♦ ❝➳❝
(−1 ≤ x ≤ 1) ✈➭ x(y) ≡ ±1, (−1 ≤ y ≤ 1)✳

❍Ö t❤ø❝

♥❣❤✐Ö♠

y(x) ≡ ±1,


✶✳✸ P❤➢➡♥❣ ♣❤➳♣ ❣✐➯✐ ♠ét sè ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝✃♣ ✶

✶✳✸✳✷

✶✾

P❤➢➡♥❣ tr×♥❤ t❤✉➬♥ ♥❤✃t

❛✳ ❍➭♠

f (x, y) ❣ä✐
f (tx, ty) = tk f (x, y)✳

➜Þ♥❤ ♥❣❤Ü❛ ✶✳✸✳✶✳

❧➭ ❤➭♠ t❤✉➬♥ ♥❤✃t ❜❐❝

k

♥Õ✉ ✈í✐

t

❜✃t ❦ú t❤×

P❤➢➡♥❣ tr×♥❤

M (x, y)dx + N (x, y)dy = 0
➤➢î❝ ❣ä✐ ❧➭ ♣❤➢➡♥❣ tr×♥❤ t❤✉➬♥ ♥❤✃t ♥Õ✉

✭✶✳✶✺✮

M (x, y), N (x, y)

❧➭ ♥❤÷♥❣ ❤➭♠

t❤✉➬♥ ♥❤✃t ❝ï♥❣ ❜❐❝

❚õ ➤Þ♥❤ ♥❣❤Ü❛ s✉② r❛ ♣❤➢➡♥❣ tr×♥❤
t❤✉➬♥ ♥❤✃t ♥Õ✉
❜✳

f (x, y)

dy
= f (x, y)
dx

❧➭ ♣❤➢➡♥❣ tr×♥❤

❧➭ ❤➭♠ t❤✉➬♥ ♥❤✃t ❜❐❝ ✵✳

P❤➢➡♥❣ ♣❤➳♣ ❣✐➯✐✳

P❤➢➡♥❣ tr×♥❤ t❤✉➬♥ ♥❤✃t ❝ã t❤Ó ➤➢❛ ✈Ò

♣❤➢➡♥❣ tr×♥❤ ♣❤➞♥ ❧✐ ❜✐Õ♥ sè ❜➺♥❣ ❝➳❝❤ ➤➷t

y = xz ✳

❚❤❐t ✈❐②✱

t❛ ❝ã t❤Ó ✈✐Õt

M (x, y) = xk M (1, y/x);
❉♦

y = xz

♥➟♥

dy = xdz + zdx✱

N (x, y) = xk N (1, y/x).

t❛ ➤➢❛ ♣❤➢➡♥❣ tr×♥❤ ✈Ò ❞➵♥❣

xk M (1, z)dx + xk N (1, z)(xdz + zdx) = 0
❤❛② ✭❣✐➯ t❤✐Õt

x = 0✮
(M (1, z) + zN (1, z))dx + xN (1, z)dz = 0.

●✐➯ sö

M (1, z) + zN (1, z) = 0✳

❈❤✐❛ ❤❛✐ ✈Õ ❝ñ❛ ♣❤➢➡♥❣ tr×♥❤ ❝❤♦

❜✐Ó✉ t❤ø❝ ♥➭② t❛ ➤➢î❝ ♣❤➢➡♥❣ tr×♥❤ ♣❤➞♥ ❧✐ ❜✐Õ♥ sè

dx
N (1, z)
+
dz = 0.
x
M (1, z) + zN (1, z)
❚Ý❝❤ ♣❤➞♥ tæ♥❣ q✉➳t ❝ã ❞➵♥❣

ln |x| +

✭✶✳✶✻✮

N (1, z)
dz = ln C1 , (C1 > 0)
M (1, z) + zN (1, z)


✷✵

P❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝✃♣ ✶

❤❛②

x = Ce−

N (1,z)
dz
M (1,z)+zN (1,z)

, C = ±C1 .

❑Ý ❤✐Ö✉

ψ(z) = −
❚❤❛②

z = y/x

N (1, z)
dz,
M (1, z) + zN (1, z)

t❛ ❝ã

x = Ceψ(z) .

t❛ ❝ã tÝ❝❤ ♣❤➞♥ tæ♥❣ q✉➳t ❝ñ❛ ♣❤➢➡♥❣ tr×♥❤ t❤✉➬♥

♥❤✃t ❝ã ❞➵♥❣
y

x = Ceψ( x ) .
❳Ðt tr➢ê♥❣ ❤î♣

M (1, z) + zN (1, z) = 0✳

●✐➯ sö

♥❣❤✐Ö♠ ❝ñ❛ ♣❤➢➡♥❣ tr×♥❤ ♥➭②✳ ❑❤✐ ➤ã ❞Ô t❤✃②

z=a

z = a

❧➭ ♠ét

❧➭ ♥❣❤✐Ö♠ ❝ñ❛

♣❤➢➡♥❣ tr×♥❤ ✭✶✳✶✻✮✱ ❞♦ ➤ã ❤➭♠
tr×♥❤ t❤✉➬♥ ♥❤✃t ❜❛♥ ➤➬✉✳

y = ax ❧➭ ♠ét ♥❣❤✐Ö♠ ❝ñ❛ ♣❤➢➡♥❣
◆❣♦➭✐ r❛ x = 0 ❝ò♥❣ ❧➭ ♠ét ♥❣❤✐Ö♠ ❝ñ❛

♣❤➢➡♥❣ tr×♥❤ ✭✶✳✶✺✮✳
❱Ý ❞ô ✶✳ ❳Ðt ♣❤➢➡♥❣ tr×♥❤

y =

y
.
x

❚r➢í❝ ❤Õt ♥❤❐♥ t❤✃②

x, y ♣❤➯✐ ❝ï♥❣ ❞✃✉✳ ➜➷t y = xz t❛ ➤➢❛


xz + z = z ✳ ❱í✐ ❣✐➯ t❤✐Õt x = 0, z − z = 0,

♣❤➢➡♥❣

tr×♥❤ ✈Ò ❞➵♥❣

♣❤➢➡♥❣

tr×♥❤ ➤➢î❝ ➤➢❛ ✈Ò ❞➵♥❣ ♣❤➞♥ ❧✐ ❜✐Õ♥ sè

dx
dz
√ = 0.
+
x
z− z
❚õ ➤ã t❛ ❝ã

❚rë ❧➵✐ ❜✐Õ♥


( z − 1)2 |x| = C1 .
y

s✉② r❛

(

y
− 1)2 |x| = C1 .
x

❙❛✉ ❦❤✐ ❣✐➯♥ ➢í❝ t❛ ➤➢î❝



y−


x=C

♥Õ✉

x > 0, y > 0✳


✶✳✸ P❤➢➡♥❣ ♣❤➳♣ ❣✐➯✐ ♠ét sè ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝✃♣ ✶

✷✶


−y − −x = C ♥Õ✉ x < 0, y < 0✳

✭C = ± C1 ✮✳

❳Ðt tr➢ê♥❣ ❤î♣ z − z = 0 t❛ ❝ã ✷ ♥❣❤✐Ö♠ z = 0, z = 1✱ t➢➡♥❣
✈í✐ ✷ ♥❣❤✐Ö♠ ❝ñ❛ ♣❤➢➡♥❣ tr×♥❤ ❜❛♥ ➤➬✉ ❧➭ y = 0, y = x(x = 0).


ø♥❣

❱Ý ❞ô ✷✳ ●✐➯✐ ♣❤➢➡♥❣ tr×♥❤

(x2 + 2xy − y 2 )dx + (y 2 + 2xy − x2 )dy = 0.
➜➷t

y = xz

t❛ ❝ã

dy = xdz + zdx✳

❚❤❛② ✈➭♦ ♣❤➢➡♥❣ tr×♥❤ ➤➲ ❝❤♦ t❛

➤➢î❝

(x2 + 2zx2 − z 2 x2 )dx + (z 2 x2 + 2x2 z − x2 )(zdx + xdz) = 0
❤❛②

(z 3 + z 2 + z + 1)dx + (z 2 + 2z − 1)xdz = 0.
❱í✐

z = −1✱

♣❤➞♥ ❧② ❜✐Õ♥ sè ✈➭ tÝ❝❤ ♣❤➞♥ t❛ ➤➢î❝

ln |x| − ln |z + 1| + ln |z 2 + 1| = ln |C1 |
❚õ ➤ã ✈í✐

C = ±C1 ✱

♥❣❤✐Ö♠ ❝ã ❞➵♥❣

x(z 2 + 1)
=C
z+1
❚rë ❧➵✐ ❜✐Õ♥ ❝ò t❛ ➤➢î❝
◆❣♦➭✐ r❛ ✈í✐

✶✳✸✳✸

z = −1✱

x2 + y 2 − C(x + y) = 0

♣❤➢➡♥❣ tr×♥❤ ❝ß♥ ❝ã ♥❣❤✐Ö♠

x + y = 0✳

P❤➢➡♥❣ tr×♥❤ q✉② ➤➢î❝ ✈Ò ♣❤➢➡♥❣ tr×♥❤ t❤✉➬♥ ♥❤✃t

❳Ðt ♣❤➢➡♥❣ tr×♥❤

dy
a1 x + b 1 y + c 1
= f(
).
dx
a2 x + b 2 y + c 2
◆Õ✉

c1 = c2 = 0

✭✶✳✶✼✮

t❤× ♣❤➢➡♥❣ tr×♥❤ tr➟♥ ❧➭ ♣❤➢➡♥❣ tr×♥❤ t❤✉➬♥ ♥❤✃t✳

❇➞② ❣✐ê ❣✐➯ sö ♠ét tr♦♥❣ ❤❛✐ sè

c1 , c2

❦❤➳❝ ✵✳ ❚❛ t×♠ ❝➳❝❤ ➤➢❛

♣❤➢➡♥❣ tr×♥❤ tr➟♥ ✈Ò ♣❤➢➡♥❣ tr×♥❤ t❤✉➬♥ ♥❤✃t ❜➺♥❣ ❝➳❝❤ ➤æ✐ ❜✐Õ♥✳


✷✷

P❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝✃♣ ✶

❛✮ ●✐➯ sö ➤Þ♥❤ t❤ø❝

a1 b 1
a2 b 2

= 0.

❉ï♥❣ ♣❤Ð♣ t❤Õ ❜✐Õ♥

❚r♦♥❣ ➤ã

u, v


x

=u+α

y

=v+β

❧➭ ❝➳❝ ❜✐Õ♥ ♠í✐✱

α, β

❧➭ ❝➳❝ sè ❝➬♥ t×♠ ➤Ó ➤➢❛

♣❤➢➡♥❣ tr×♥❤ ✭✶✳✶✼✮ ✈Ò ♣❤➢➡♥❣ tr×♥❤ t❤✉➬♥ ♥❤✃t✳ ❚❤❛② ❜✐Õ♥

u, v

tr♦♥❣ ♣❤➢➡♥❣ tr×♥❤ ✭✶✳✶✼✮ t❛ ❝ã

a1 u + b1 v + a1 α + b1 β + c1
dv
= f(
).
du
a2 u + b2 v + a2 α + b2 β + c2
➜Ó ✭✶✳✶✽✮ ❧➭ ♣❤➢➡♥❣ tr×♥❤ t❤✉➬♥ ♥❤✃t t❤× ❤➭♠
❜❐❝ ✵ ✈í✐

u, v

❤❛②

f (tu, tv) = f (u, v)✳

f

✭✶✳✶✽✮

♣❤➯✐ ❧➭ t❤✉➬♥ ♥❤✃t

❉♦ ➤ã t❛ ❝❤Ø ❝➬♥ t×♠

α, β

t❤á❛

♠➲♥ ❤Ö ♣❤➢➡♥❣ tr×♥❤


a α + b β + c
1
1
1
a2 α + b2 β + c2

=0
=0

❍Ö ♥➭② ❧✉➠♥ ❝ã ♥❣❤✐Ö♠ ❞✉② ♥❤✃t ✈× ➤Þ♥❤ t❤ø❝ ❈r❛♠❡ ❝ñ❛ ♥ã ❦❤➳❝ ✵✳
❜✮ ❚r➢ê♥❣ ❤î♣ ➤Þ♥❤ t❤ø❝

a1 b 1
a2 b 2

= 0.

❑❤✐ ➤ã ❤❛✐ ❞ß♥❣ ❝ñ❛ ➤Þ♥❤ t❤ø❝ tØ ❧Ö tø❝ ❧➭ tå♥ t➵✐ ♠ét sè t❤ù❝
❝❤♦

a2 = λa1 , b2 = λb1 ✳

❉♦ ➤ã ♣❤➢➡♥❣ tr×♥❤ ✭✶✳✶✼✮ ❝ã ❞➵♥❣

dy
a1 x + b 1 y + c 1
= f(
) = ϕ(z)
dx
λ(a1 x + b1 y) + c2
❉♦

dz
dy
= a1 + b 1
dx
dx

(✈í✐ z = a1 x + b1 y).

♥➟♥ t❛ ➤➢î❝ ♣❤➢➡♥❣ tr×♥❤ ♣❤➞♥ ❧✐ ❜✐Õ♥ sè

dz
= a1 + b1 ϕ(z).
dx

λ s❛♦


✶✳✸ P❤➢➡♥❣ ♣❤➳♣ ❣✐➯✐ ♠ét sè ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝✃♣ ✶

✷✸

❱Ý ❞ô ✶✳ ●✐➯✐ ♣❤➢➡♥❣ tr×♥❤

dy
−7x + 3y + 7
=
.
dx
3x − 7y − 3
➜➷t

x = u + α, y = v + β tr♦♥❣ ➤ã α, β ❧➭ ♥❣❤✐Ö♠

−7α + 3β + 7 = 0
3α − 7β − 3
=0

❚❛ t×♠ ➤➢î❝

α = 1, β = 0✳

❝ñ❛ ❤Ö s❛✉

❙❛✉ ♣❤Ð♣ t❤Õ ❜✐Õ♥ tr➟♥ ♣❤➢➡♥❣ tr×♥❤ ➤➢❛

✈Ò ❞➵♥❣

dv
−7u + 3v
−7 + 3v/u
=
=
.
du
3u − 7v
3 − 7v/u
➜➷t

v = zu

t❛ ➤➢❛ ✈Ò ♣❤➢➡♥❣ tr×♥❤

z+u

−7 + 3z
dz
=
.
du
3 − 7z

y
v
=
t❛ t×♠ ➤➢î❝ tÝ❝❤
u
x−1
♣❤➞♥ tæ♥❣ q✉➳t ❝ñ❛ ♣❤➢➡♥❣ tr×♥❤ ❜❛♥ ➤➬✉ ❧➭ |y+x−1|5 |y−x+1|2 = C.
❚Ý❝❤ ♣❤➞♥ ♣❤➢➡♥❣ tr×♥❤ ♥➭② ✈➭ ❝❤ó ý

z=

❱Ý ❞ô ✷✳ ❳Ðt ♣❤➢➡♥❣ tr×♥❤

(x + y − 2)dx + (x − y + 4)dy = 0.
❉ï♥❣ ♣❤Ð♣ ➤æ✐ ❜✐Õ♥


x = u − 1
y = v + 3
❚❛ ➤➢î❝ ♣❤➢➡♥❣ tr×♥❤
➜➞②
2

❧➭

♣❤➢➡♥❣
2

u + 2uv − v = C ✳

(u + v)du + (u − v)dv = 0✳

tr×♥❤

t❤✉➬♥

❚❤❛② ❧➵✐ ❜✐Õ♥

♥❤✃t✱

x, y

tÝ❝❤

♣❤➞♥

♥ã

t❛

➤➢î❝

t❛ ➤➢î❝ tÝ❝❤ ♣❤➞♥ tæ♥❣ q✉➳t

❝ñ❛ ♣❤➢➡♥❣ tr×♥❤ ➤➲ ❝❤♦ ❧➭

x2 + 2xy − y 2 − 4x + 8y = C.


✷✹

P❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❝✃♣ ✶

✶✳✸✳✹

P❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ t♦➭♥ ♣❤➬♥✳ ❚❤õ❛ sè tÝ❝❤ ♣❤➞♥

➜Þ♥❤ ♥❣❤Ü❛ ✶✳✸✳✷✳

P❤➢➡♥❣ tr×♥❤

M (x, y)dx + N (x, y)dy = 0

✭✶✳✶✾✮

❣ä✐ ❧➭ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ t♦➭♥ ♣❤➬♥ ♥Õ✉ tå♥ t➵✐ ❤➭♠

U (x, y)

❦❤➯ ✈✐ s❛♦

❝❤♦ ✈✐ ♣❤➞♥ t♦➭♥ ♣❤➬♥ ❝ñ❛ ♥ã

dU (x, y) = M (x, y)dx + N (x, y)dy.
❚❛ ❧✉➠♥ ❣✐➯ t❤✐Õt r➺♥❣ ❝➳❝ ❤➭♠ sè
❤➭♠ r✐➟♥❣

∂M ∂N
,
∂y ∂x

M (x, y), N (x, y) ❝ï♥❣ ✈í✐ ❝➳❝ ➤➵♦

❧✐➟♥ tô❝ tr♦♥❣ ♠ét ♠✐Ò♥ ➤➡♥ ❧✐➟♥

G

♥➭♦ ➤ã✳

◆❤➢ ✈❐② ♥Õ✉ ✭✶✳✶✾✮ ❧➭ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ t♦➭♥ ♣❤➬♥ t❤× t❛ ❝ã

dU (x, y) = 0

✈➭ ❞♦ ➤ã

U (x, y) = C

❧➭ tÝ❝❤ ♣❤➞♥ tæ♥❣ q✉➳t ❝ñ❛ ♥ã✳

❱✃♥ ➤Ò ➤➷t r❛ ❧➭ ❦❤✐ ♥➭♦ ♣❤➢➡♥❣ tr×♥❤ ✭✶✳✶✾✮ ❧➭ ♣❤➢➡♥❣ tr×♥❤ ✈✐
♣❤➞♥ t♦➭♥ ♣❤➬♥✱ ✈➭ ♥Õ✉ ❧➭ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ t♦➭♥ ♣❤➬♥ t❤× t×♠
❤➭♠ sè

U (x, y)

♥❤➢ t❤Õ ♥➭♦✳ ➜Þ♥❤ ❧Ý s❛✉ ❧➭ ❝➞✉ tr➯ ❧ê✐ ❝❤♦ ❤❛✐ ❝➞✉

❤á✐ tr➟♥✳
➜Þ♥❤ ❧Ý ✶✳✸✳✸✳

❧✐➟♥

➜Ó ✭✶✳✶✾✮ ❧➭ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ t♦➭♥ ♣❤➬♥ tr♦♥❣ ♠✐Ò♥ ➤➡♥

G t❤× ➤✐Ò✉ ❦✐Ö♥ ❝➬♥ ✈➭ ➤ñ ❧➭
∂N
∂M
=
,
∂y
∂x

∀(x, y) ∈ G.

(x0 , y0 ) ∈ G ❜✃t ❦ú s❛♦ ❝❤♦ M (x, y), N (x, y) ❦❤➠♥❣ ➤å♥❣
tr✐Öt t✐➟✉✱ ❤➭♠ sè U (x, y) ➤➢î❝ tÝ♥❤ t❤❡♦ ♠ét tr♦♥❣ ❤❛✐ ❝➠♥❣ t❤ø❝ s❛✉

❑❤✐ ➤ã ✈í✐

y

x

U (x, y) =

M (x, y0 )dx +
x0

N (x0 , y)dy,

(N (x0 , y) = 0).

y

M (x, y)dx +
x0

(M (x, y0 ) = 0).

y0

x

U (x, y) =

N (x, y)dy,

y0

t❤ê✐


P ột số trì



ứ ề ệ sử trì

t ó ớ ọ

x, y G

t ó

M (x, y)dx + N (x, y)dy = dU (x, y) =

U
U
dx +
dy.
x
y

ừ s r

M (x, y) =
ì tr ề
tứ

U
,
x

N (x, y) =

U
.
y

U U
,
tồ t tụ
x y
tr t y x t ứ t ó
G,

M
2U
=
,
y
yx

ế ủ

N
2U
=
.
x
xy

tết ế tr ủ tứ tr tụ tr

G

ó ỗ ợ

2U
,
yx

2U
xy

tụ ú



N
M
=
y
x
ề ệ ủ sử tr ề

(x, y) G.
G

t ó

M
N
=
.
y
x
sẽ tì

U (x, y) tỏ ề ệ tr ị ĩ

trì t rớ ết ò ỏ

U
= M (x, y).
x
ó

x

U (x, y) =

M (x, y)dx + (y).
x0


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