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
ủ
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
ứ
x
ế t tr
tể ự
x
yn (x) = y0 +
f (, yn1 ( )d.
x0
P trì
ớ
|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
sè
♣❤➢➡♥❣
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