Phép biến đổi laplace và ứng dụng giải phương trình vi phân

❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖
✣❸■ ❍➴❈ ✣⑨ ◆➂◆●

✖✖✖✖✖

❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P

P❍➆P ❇■➌◆ ✣✃■ ▲❆P▲❆❈❊
❱⑨ Ù◆● ❉Ư◆● ●■❷■ P❍×❒◆● ❚❘➐◆❍ ❱■ P❍❹◆

❙✐♥❤ ✈✐➯♥ t❤ü❝ ❤✐➺♥✿ ◆❣✉②➵♥ ❚❤à ▲➺ ❍➡♥❣
●✐→♦ ✈✐➯♥ ữợ r




▼ư❝ ❧ư❝
▼Ð ✣❺❯
✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚
✶✳✶
✶✳✷
✶✳✸

✸
✺

P❤÷ì♥❣ ♣❤→♣ ❦❤❛✐ tr✐➸♥ t❤ø❛ sè r✐➯♥❣ ♣❤➛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❚➼❝❤ ♣❤➙♥ s✉② rë♥❣ ✈➔ sü ❤ë✐ tö

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

◆❤➢❝ ❧↕✐ ♠ët sè ❦❤→✐ ♥✐➺♠ tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥

✳ ✳

✷ P❍➆P ❇■➌◆ ✣✃■ ▲❆P▲❆❈❊

✺
✻
✶✸

✶✺

✷✳✶

P❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✺

✷✳✷

P❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ♥❣÷đ❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳





ỵ t ❝❤➟♣

✷✸

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

✸ Ù◆● ❉Ư◆● P❍➆P ❇■➌◆ ✣✃■ ▲❆P▲❆❈❊ ●■❷■ P❍×❒◆●
❚❘➐◆❍ ❱■ P❍❹◆
✷✻
✸✳✶

Ù♥❣ ❞ư♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥

✸✳✷

Ù♥❣ ❞ư♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❝❤♦ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✸✵

✸✳✸

◆❤ú♥❣ ÷✉ ✤✐➸♠ ✈➔ ♥❤÷đ❝ ✤✐➸♠ ❝õ❛ ✈✐➺❝ →♣ ❞ư♥❣ ♣❤➨♣ ❜✐➳♥
✤ê✐ ▲❛♣❧❛❝❡ tr♦♥❣ ✈✐➺❝ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳

❑➌❚ ▲❯❾◆
❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖

✷✻

✸✸

✸✹
✸✺

✷

é
ỵ ỹ ồ t
t➼❝❤ ❚♦→♥ ❤å❝ ✤➣ ❝â sü ❜✐➳♥ ✤ê✐ ♠↕♥❤ ♠➩✱ tr♦♥❣ ✤â ❧➽♥❤
✈ü❝ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❦❤ỉ♥❣ ♥❣ø♥❣ ✤÷đ❝ ♣❤→t tr✐➸♥ ✈➻ ♥â ❝â r➜t ♥❤✐➲✉
ù♥❣ ❞ö♥❣ t❤ü❝ t✐➵♥✳ ❱➻ t❤➳✱ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ♥❤✐➲✉ ♣❤÷ì♥❣
♣❤→♣ ✤➸ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ♥❤÷ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r✱ ♣❤÷ì♥❣
♣❤→♣ ❝❤✉é✐ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✱✳✳✳ ❤❛② ù♥❣ ❞ư♥❣ t✐♥ ❤å❝✳ ❚r♦♥❣ sè
✤â✱ ♣❤÷ì♥❣ ♣❤→♣ ✈➟♥ ❞ư♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ✤➸ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐
♣❤➙♥ ✤÷đ❝ ữỡ õ ỵ ♥❣❤➽❛✳ ❱ỵ✐ ♠♦♥❣
♠✉è♥ ❝â t❤➸ ❤✐➸✉ ❦➽ ❤ì♥ ✈➲ ❝→❝ ❞↕♥❣ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤✱
❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ũ ợ sỹ ủ ỵ t t ữợ ❝õ❛ t❤➛②
❣✐→♦ ❚❙ ▲➯ ❍↔✐ ❚r✉♥❣ ♥➯♥ ❡♠ ✤➣ ❝❤å♥ ✤➲ t➔✐ ✏P❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ✈➔

✷✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉✿

ù♥❣ ❞ư♥❣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✑ ✤➸ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ tèt ♥❣❤✐➺♣✳

❚❤ü❝ ❤✐➺♥ ✤➲ t➔✐ ✏P❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ✈➔ ù♥❣ ❞ư♥❣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤
✈✐ ♣❤➙♥✑✱ t→❝ ❣✐↔ ữợ ử r t ❝➟♥✱ t➻♠
❤✐➸✉ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ♠ët ✈➜♥ ✤➲ ❚♦→♥ ❤å❝ ❝á♥ ❦❤→ ♠ỵ✐ ✤è✐ ✈ỵ✐ ❜↔♥ t❤➙♥✳ ❚ø
✤â✱ ❤➻♥❤ t❤➔♥❤ ❦❤↔ ♥➠♥❣ tr➻♥❤ ❜➔② ♠ët ✈➜♥ ✤➲ t♦→♥ ❤å❝ trø✉ t÷đ♥❣ ♠ët
❝→❝❤ ❧♦❣✐❝ ✈➔ ❝â ❤➺ t❤è♥❣✳ ▲✉➟♥ ✈➠♥ ♥❤➡♠ ự ỳ ợ ữỡ
tr õ t ự ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ✤➸ ❣✐↔✐
tr➯♥ ❝ì s tờ ủ ỵ t ❝❤➜t ❝õ❛ ♣❤➨♣ ❜✐➳♥ ✤è✐
▲❛♣❧❛❝❡ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✳ ❚❤ü❝ ❤✐➺♥ ❜➔✐ ❧✉➟♥ ✈➠♥ ♥➔②✱ t→❝ ❣✐↔ ♠✉è♥
❝õ♥❣ ❝è ✈➔ ❤➺ t❤è♥❣ ❧↕✐ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ✈➲ t➼❝❤ ♣❤➙♥ s rở ữỡ
tr q ợ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ ♠ët ✈➜♥ ✤➲ ❝õ❛

✸✳ ❇è ❝ö❝ ❝õ❛ ❧✉➟♥ ✈➠♥✿

t♦→♥ ❤å❝✳

◆❣♦➔✐ ♣❤➛♥ ♠ð ✤➛✉✱ ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✈➔ ❦➳t ❧✉➟♥✱ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛

✸


❧➔♠ ❤❛✐ ♣❤➛♥
P❤➛♥ ✶✿ P❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡
P❤➛♥ ♥➔② s➩ tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡✱ ♣❤➨♣ ❜✐➳♥
✤ê✐ ▲❛♣❧❛❝❡ ✈➔ ♥❤ú♥❣ t➼♥❤ ❝❤➜t ❝õ❛ ❝❤ó♥❣ ✤➸ ❧➔♠ ❝ì sð ❝❤♦ ♣❤➛♥ s❛✉ ❧➔
♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥✳
P❤➛♥ ✷✿ ⑩♣ ❞ö♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ✤➸ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥
P❤➛♥ ♥➔② tr➻♥❤ ❜➔② ❝→❝❤ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤
✈✐ ♣❤➙♥ ❜➡♥❣ ✈✐➺❝ →♣ ử ờ rỗ ũ ử ✤➸
♠✐♥❤ ❤å❛ rã ❤ì♥ ✈➲ ❝→❝❤ ❣✐↔✐ ♥➔②✳

✹


❈❤÷ì♥❣ ✶

❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝→❝ ❦✐➳♥ t❤ù❝ ✤÷đ❝ t❤❛♠ ❦❤↔♦ tr♦♥❣ ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪✱
❬✷❪✱ ❬✸❪✱ ❬✹❪✳

✶✳✶ P❤÷ì♥❣ ♣❤→♣ ❦❤❛✐ tr✐➸♥ t❤ø❛ sè r✐➯♥❣ ♣❤➛♥
❈❤♦ ♣❤➙♥ t❤ù❝

x

❜➟❝

m

✈➔

n

t÷ì♥❣ ù♥❣✱

❑❤❛✐ tr✐➸♥

u(x)
✱ tr♦♥❣ ✤â u(x) ✈➔ v(x)
v(x)
✈ỵ✐ m < n✱ t❛ ❧➔♠ ♥❤÷ s❛✉✿

f (x) =

v(x)

❧➔ ❝→❝ ✤❛ t❤ù❝ ❝õ❛

t❤➔♥❤ ❝→❝ t❤ø❛ sè ✤ì♥ ❣✐↔♥ ❝â ❞↕♥❣

v(x) = (x − x1 )k1 (x − x2 )k2 ...(x − xr )kr

tr♦♥❣ ✤â

k1 + k2 + ... + kr = n.

◆❤÷ ✈➟②✱ ❝â t❤➸ ❦❤❛✐ tr✐➸♥ ❤➔♠
sì ❝➜♣ ❝â ❞↕♥❣✿

Aij
✱
(x − xi )ki −j+1

f (x) =

tr♦♥❣ ✤â

i

u(x)
v(x)

t❤➔♥❤ tê♥❣ ❝→❝ ♣❤➙♥ sè

❧➜② t➜t ❝↔ ❝→❝ ❣✐→ trà tø

1

✤➳♥

r✱

1 ✤➳♥ ki ✳
Aij

❚❛ ❝â F (x) =
✱ t➜t ❝↔ ❤➺ sè Aij ❝õ❛ ❦❤❛✐ tr✐➸♥
ki −j+1
i=1 j=1 (x − xi )
1
dj−1
✤÷đ❝ t➻♠ t❤❡♦ ❝ỉ♥❣ t❤ù❝ Aij =
lim { j−1 [(x − x1 )ki F (x)]}.
(j − 1)! x→x1 dx

❝á♥ ❥ ❧➜② t➜t ❝↔ ❝→❝ ❣✐→ trà sè tø

r

♥➔②

ki

❚❤❛② ❝❤♦ ❝ỉ♥❣ t❤ù❝ ♥➔② ❝â t❤➸ ❞ị♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ sì ❝➜♣ tr♦♥❣
♣❤➨♣ t➼♥❤ t➼❝❤ ♣❤➙♥ ❦❤✐ t➼♥❤ t➼❝❤ ♣❤➙♥ ❝→❝ ♣❤➙♥ sè ❤ú✉ t✛✳ ✣➦❝ ❜✐➺t ✤✐➲✉
♥➔② r➜t t❤✉➟♥ ❧đ✐ tr♦♥❣ ❝→❝ tr÷í♥❣ ❤đ♣ t➜t ❝↔ ❝→❝ ♥❣❤✐➺♠ ♣❤ù❝ ❝õ❛ ♠➝✉ sè

v(x)

✤ì♥ ✈➔ ✤ỉ✐ ♠ët ❧✐➯♥ ❤ñ♣✳

✺


v(x)

◆➳✉ t➜t ❝↔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛

✤ì♥ t❤➻ ❦❤❛✐ tr✐➸♥ ❝â ❞↕♥❣ ✤ì♥ ❣✐↔♥✿

n

v(x) = (x − x1 )(x − x2 )...(x − xn )xj = xk khi j = k ✱ F (x) =
tr♦♥❣ ✤â

Aj =

❱➼ ❞ö ✶✳✶✳

Aj
✱
j=1 x − xj

u(xj )
✳
v (xj )

P❤➙♥ t➼❝❤ ♣❤➙♥ t❤ù❝

F (x) =

8 − (x + 2)(4x + 10)
(x + 1)(x + 2)2

t❤➔♥❤ tê♥❣

❝→❝ ♣❤➙♥ t❤ù❝ ✤ì♥ ❣✐↔♥✳

●✐↔✐✿

F (x) t❤➔♥❤✿
A
B
C
F (x) =
+
+
x + 1 x + 2 (x + 2)2
A(x + 2)2 + B(x + 1)(x + 2) + C(x + 1)
=
(x + 1)(x + 2)

❚❛ ❜✐➳♥ ✤ê✐

❚❤❡♦ ❜✐➸✉ t❤ù❝ ✤➣ ❝❤♦ t❛ ❝â ✿

A(x + 2)2 + B(x + 1)(x + 2) + C(x + 1) = 8 − (x + 2)(4x + 10).

❈❤♦
❈❤♦
❈❤♦

x = −2 t❛ ❝â C = −8.
x = −1 t❛ ❝â A = 2.
x = 0 t❛ ❝â B = −6.

❉♦ ✤â✿

F (x) =

2
6
8
−
−
.
x + 1 x + 2 (x + 2)2

✶✳✷ ❚➼❝❤ ♣❤➙♥ s✉② rë♥❣ ✈➔ sü ❤ë✐ tö
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ✭❚➼❝❤ ♣❤➙♥ s✉② rë♥❣✮

●✐↔ sû

tr➯♥ ❦❤♦↔♥❣

[a, +∞)

✈➔ ❦❤↔ t➼❝❤ tr➯♥ ♠å✐ ✤♦↕♥

f ❧➔ ♠ët ❤➔♠ sè ①→❝
[a, b] ✈ỵ✐ b > a. ◆➳✉

✤à♥❤

b

lim

f (x)dx = I,

b→+∞
a
tr♦♥❣ ✤â
rë♥❣ ❝õ❛

I ∈ R✱ I = +∞ ❤♦➦❝ I = −∞ t❤➻ I ✤÷đ❝
f tr➯♥ ❦❤♦↔♥❣ [a, +∞) ✈➔ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔
+∞

f (x)dx.
a
✻

❣å✐ ❧➔ t➼❝❤ ♣❤➙♥ s✉②


+

f (x)dx

õ t õ r

tỗ t

I

ỳ tù❝ ❧➔

I ∈R

t❤➻

a
+∞

f (x)dx

t❛ ♥â✐ r➡♥❣

❧➔ ❤ë✐ tö✳ ❚➼❝❤ ♣❤➙♥ ❦❤ỉ♥❣ ❤ë✐ tư ❣å✐ ❧➔ ♣❤➙♥ ❦ý✳

a

❱➼ ❞ư ✶✳✷✳

+∞

e−x dx.

❚➼♥❤

0
❱ỵ✐ ♠å✐ sè t❤ü❝ ❜ ❃ ✵✱ t❛ ❝â✿

b

e−x dx = (−e−x ) |b0 = 1 − e−b
0
b

e−x dx = lim (1 − e−b ) = 1.

lim

b→+∞

b→+∞

0
+∞

e−x dx

❉♦ ✤â

❤ë✐ tö ✈➔

0
+∞

e−x dx = 1.
0

❱➼ ❞ö ✶✳✸✳

+∞

b

dx
= lim
1 + x2 b→+∞
0

❱➼ ❞ö ✶✳✹✳

dx
π
=
lim
arctan
b
=
.
1 + x2 b→+∞
2
0

+∞

b

dx = lim b = +∞.

dx = lim

b+

0

b+

0
+

dx

õ t s rở

tỗ t ữ

0

ỵ

+

f (x)dx

t s rở

a

+

g(x)dx



a

ở


+∞

[f (x) + g(x)]dx

tư t❤➻ t➼❝❤ ♣❤➙♥ s✉② rë♥❣

❝ơ♥❣ ❤ë✐ tö ✈➔

a
+∞

+∞

[f (x) + g(x)]dx =
a

+∞

f (x)dx +

g(x)dx.

a

a

+∞

f (x)dx

❜✮ ◆➳✉ t➼❝❤ ♣❤➙♥

λ

❤ë✐ tö ✈➔

❧➔ ♠ët ❤➡♥❣ sè t❤ü❝

a
+∞

+∞

λf (x)dx

t❤➻ t➼❝❤

ỵ

f (x)dx =

ở tử

a

+

a

f (x)dx.
a

f ♠ët ❤➔♠ sè ①→❝ ✤à♥❤ tr➯♥
[a, b) ✈ỵ✐ b > a✳ ◆➳✉ f (x) ≥ 0 ✈ỵ✐

●✐↔ sû

❦❤↔ t➼❝❤ tr➯♥ ♠å✐ ✤♦↕♥

[a, +∞)✱
x ∈ [a, +∞)

❦❤♦↔♥❣
♠å✐

+∞

f (x)dx

t❤➻ t➼❝❤ ♣❤➙♥

❧✉æ♥ ❧✉æ♥ tỗ t ỳ

+

a
b

ự

t

f (x)dx b a✳

F (b) =

b ≥b

◆➳✉

t❤➻✿

a
b

b

f (x)dx =

F (b ) =
a

t➠♥❣ tr➯♥

b

lim

[a, +)

tỗ t

f (x)dx

tự tỗ t

a



a
+

b

f (x)dx =
a

sup F (b) =

b∈[a,+∞)

sup

f (x)dx.

b∈[a,+∞)

a

+∞

f (x)dx

❍✐➸♥ ♥❤✐➯♥

b
❤ë✐ tö ❦❤✐ ✈➔ ❝❤➾ ❦❤✐

b

b→

sup
b≥a

a
❤➔♠ sè

lim F (b) =

b→+∞

+∞

f (x)dx

b→+∞

b

b

b → F (b)

f (x)dx ≥ F (b).

f (x)dx = F (b) +

f (x)dx +
a

❍➔♠ sè

b

b

f (x)dx

❜à ❝❤➦♥ tr➯♥

[a, +∞)✳

a

✽

f (x)dx < +∞✱
a

tù❝ ❧➔


ỵ s s

sỷ

f



[a, +) ❦❤↔ t➼❝❤ tr➯♥ ♠å✐ ✤♦↕♥
0 ≤ f (x) ≤ g(x) ✈ỵ✐ ♠å✐ x ∈ [a, +∞) t❤➻

tr➯♥ ❦❤♦↔♥❣
◆➳✉✿

+∞

g ❧➔ ❤❛✐ ❤➔♠ sè
[a, b] ✈ỵ✐ b > a✳

①→❝ ✤à♥❤

+∞

f (x)dx ≤
a

g(x)dx.
a

❚ø ✤â s✉② r❛✿

+∞

+∞

g(x)dx

◆➳✉

a
+∞

❤ë✐ tö✳

a
+∞

f (x)dx

◆➳✉

f (x)dx

❤ë✐ tö t❤➻

❍➺ q✉↔ ✶✳✶✳

g(x)dx

♣❤➙♥ ❦➻ t❤➻

a

♣❤➙♥ ❦➻✳

a
●✐↔ sû

f

g ❧➔ ♥❤ú♥❣ ❤➔♠ sè ①→❝ ✤à♥❤ tr➯♥ ❦❤♦↔♥❣ [a, +∞)
✤♦↕♥ [a, b] ✈ỵ✐ ❜ ❃ ❛✳ ◆➳✉ f (x) ≥ 0✱ g(x) ≥ 0 tr➯♥

✈➔ ❦❤↔ t➼❝❤ tr➯♥ ♠å✐

✈➔

+∞

[a, +∞)

f ∼g

✈➔

❦❤✐

x → +∞

+∞

f (x)dx

t❤➻ ❝→❝ t➼❝❤ ♣❤➙♥

g(x)dx

✈➔

a

a

❝ị♥❣ ❤ë✐ tư ❤♦➦❝ ❝ị♥❣ ♣❤➙♥ ❦➻✳

❱➼ ❞ư ✶✳✺✳

+∞
2

e−x dx.

❳➨t t➼♥❤ ❤ë✐ tö ❝õ❛ t➼❝❤ ♣❤➙♥

0
+∞
❚❛ ❝â✿

2

0 < e−x ≤ e−x

✈ỵ✐ ♠å✐

x ≥ 1✳

e−x dx

❚❛ ❜✐➳t r➡♥❣

❤ë✐ tư✳

0
+∞
2

e−x dx

❉♦ ✤â

❤ë✐ tử

0

ú ỵ

ỵ q ữủ ử trữớ ủ

số ữợ t ♣❤➙♥ ❦❤ỉ♥❣ ➙♠ ✭✈ỵ✐ ❣✐→ trà ✤õ ❧ỵ♥ ❝õ❛ ❛✮✳

✣à♥❤ ỵ sỹ ở tử ừ t➼❝❤ ♣❤➙♥✮
♠ët ❤➔♠ sè ①→❝ ✤à♥❤ tr➯♥

[a, +∞)

✈➔ ❦❤↔ t➼❝❤ tr➯♥ ♠å✐ ✤♦↕♥

f ❧➔
[a, b]✱ b > a✳
●✐↔ sû

+∞

f (x)dx

❑❤✐ ✤â t➼❝❤ ♣❤➙♥

❤ë✐ tư ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ✈ỵ✐ ♠ët sè ❞÷ì♥❣

a

✾

ε

❜➜t


tỗ t ởt số tỹ

b0 a

s

b2

(b1 , b2 ∈ R) b2 ≥ b1 ≥ b0 ⇒ |

f (x)dx| < ε.
b1

❈❤ù♥❣ ♠✐♥❤✿

F

●å✐

❧➔ ❤➔♠ sè ①→❝ ✤à♥❤ tr➯♥

[a, +∞)

❜ð✐

b

b → F (b) =

f (x)dx.
a

❚❤❡♦ t✐➯✉ ❝❤✉➞♥ ❈❛✉❝❤② ✈➲ sü tỗ t ợ ừ số


> 0, ∃b0 ≥ a

∃ lim F (b)
b→+∞

s❛♦ ❝❤♦✿

∀b1 , b2 ∈ R : b2 ≥ b1 ≥ b0 ⇒ |F (b2 ) − F (b1 )| < ε;
b2

F (b2 ) − F (b1 ) =

♠➔

f (x)dx

♥➯♥ tø ✤â s✉② r❛ ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ✭❚➼❝❤ ♣❤➙♥ ❤ë✐ tö t✉②➺t ✤è✐✮
b1

+∞

❚❛ ♥â✐ r➡♥❣ t➼❝❤ ♣❤➙♥

+∞

f (x)dx

|f (x)|dx

❤ë✐ tö t✉②➺t ố

ỵ
a

ở tử

a
ở tử tt ố t❤➻ ❤ë✐ tö✳

+∞

❈❤ù♥❣ ♠✐♥❤✿

f (x)dx

●✐↔ sû t➼❝❤ ♣❤➙♥

❤ë✐ tö t✉②➺t ố tự t

a
+

|f (x)|dx



ở tử

>0

tũ ỵ t t sỹ

a
ở tử ừ t tỗ t

b0 ≥ a

s❛♦ ❝❤♦✿

b2

b2 ≥ b1 ≥ b0 ⇒ |

|f (x)|dx| < ε.
b1

b2
❉♦ ✤â

|

b2

f (x)dx| ≤ |
b1

|f (x)|dx| < ε

✈ỵ✐

b2 ≥ b1 ≥ b0 .

b1
+∞

f (x)dx

❱➟② t➼❝❤ ♣❤➙♥

❤ë✐ tö ✭t❤❡♦ t✐➯✉ ❝❤✉➞♥ ❈❛✉❝❤②✮✳

a
✶✵


❱➼ ❞ö ✶✳✻✳

+∞
❳➨t sü ❤ë✐ tö ❝õ❛ t➼❝❤ ♣❤➙♥

e−x sin αxdx✱ α ∈ R.

I=
0
+∞

❱➻

|e−x sin αx| ≤ e−x

✈ỵ✐ ♠å✐

x∈R

e−x dx

✈➔

❤ë✐ tö ♥➯♥ t➼❝❤ ♣❤➙♥

I

0
❤ë✐ tö t✉②➺t ✤è✐✳
❑❤✐ ①➨t sü ❤ë✐ tư ❝õ❛ ❝→❝ t➼❝❤ ♣❤➙♥ ❦❤ỉ♥❣ ❤ë✐ tư t✉②➺t ✤è✐✱ t tữớ
tợ ở tử rt

ỵ ❉✐r✐❝❤❧❡t✮
❛✮ ❍➔♠ sè

f

❧✐➯♥ tö❝ tr➯♥

●✐↔ sû

b

[a, +∞)

✈➔ ❤➔♠ sè

b → F (b) =

f (x)dx

❜à

g

❝â

a

[a, +∞) ✭ ∃M > 0 s❛♦ ❝❤♦✿ |F (b)| ≤ M ✱ ∀b ≥ a✮✱
sè g ✤ì♥ ✤✐➺✉ tr➯♥ [a, +∞) ✈➔ lim g(x) = 0✳

❝❤➦♥ tr➯♥
❜✮ ❍➔♠

x→+∞

+∞

f (x)g(x)dx

❑❤✐ ✤â t➼❝❤ ♣❤➙♥

❈❤ù♥❣ ♠✐♥❤✿

❤ë✐ tö✳

a
❚❛ ❝❤ù♥❣ ♠✐♥❤ ỵ ợ tt ờ s số

❧✐➯♥ tö❝ tr➯♥

[a, +∞)✳

❚❛ →♣ ❞ö♥❣ t✐➯✉ ❝❤✉➞♥ ❈❛✉❝❤② ✈➲ sỹ ở tử ừ t

x

tũ ỵ t r➡♥❣ ❤➔♠ sè

x → F (x) =

f (t)dt

ε>0

❧➔ ♠ët ♥❣✉②➯♥ ❤➔♠ ❝õ❛

a
❤➔♠ sè

f

tr➯♥

[a, +∞)✳

b2

❱ỵ✐ ♠å✐ sè t❤ü❝

b2 ≥ b1 ≥ a

b2

g(x)dF (x) = [F (x)g(x)] |bb21 −

f (x)g(x)dx =
b1

t❛ ❝â✿

b2

b1

F (x)g (x)dx
b1

b2

= F (b2 )g(b2 ) − F (b1 )g(b1 ) −

F (x)g (x)dx
b1

❱➻ ❤➔♠ sè

g

✤ì♥ ✤✐➺✉ tr➯♥

[a, +∞)

♥➯♥ ✤↕♦

g

ừ õ ổ ờ

tr ỵ tr tr rở ừ t
tỗ t↕✐ ♠ët sè t❤ü❝

c ∈ [b1 , b2 ]

b2

b2

g (x)dx = F (c)[g(b2 ) − g(b1 )].

F (x)g (x)dx = F (c)
b1

s❛♦ ❝❤♦✿

b1
✶✶


❚❤❛② ✈➔♦ ❝ỉ♥❣ t❤ù❝ tr➯♥ t❛ ✤÷đ❝✿

b2

f (x)g(x)dx = [F (b2 ) − F (c)]g(b2 ) + [F (c) − F (b1 )]g(b1 ).
b1

❉♦ ✤â

b2

|

f (x)g(x)dx| ≤ 2M |g(b2 )| + 2M |g(b1 )|.
b1



lim g(x) = 0

tỗ t ởt sè t❤ü❝

x→+∞

b0 ≥ a

x ≥ b0 ⇒ |g(x)| <

s❛♦ ❝❤♦

ε
.
4M

❈✉è✐ ❝ị♥❣ t❛ ✤÷đ❝

b2

b2 ≥ b1 ≥ b0 ⇒ |

f (x)g(x)dx| < 2M



+ 2M
= .
4M
4M

b1
+

f (x)g(x)dx

t

ở tử

ỵ ✭❉➜✉ ❤✐➺✉ ❆❜❡❧✮
a

❛✮ ❍➔♠ sè

f

◆➳✉

❧✐➯♥ tö❝ tr➯♥ ❦❤♦↔♥❣

+∞

[a, +∞)

f (x)dx

✈➔ t➼❝❤ ♣❤➙♥

❤ë✐ tư✱

a
❜✮ ❍➔♠ sè

g

✤ì♥ ✤✐➺✉ ✈➔ ❜à ❝❤➦♥ tr➯♥ ❦❤♦↔♥❣

[a, +∞)

+∞

f (x)g(x)dx

t❤➻ t➼❝❤ ♣❤➙♥

❤ë✐ tư✳

a

❈❤ù♥❣ ♠✐♥❤✿
sè

g

❱➻ ❤➔♠ sè

❝â ❣✐ỵ✐ ❤↕♥ ❤ú✉ ❤↕♥ ❦❤✐

❍➔♠ sè

g−A

✤ì♥ ✤✐➺✉ tr➯♥

g ✤ì♥ ✤✐➺✉ ✈➔ ❜à ❝❤➦♥ tr➯♥ [a, +∞)
x → +∞✿ lim g(x) = A ∈ R✳

♥➯♥ ❤➔♠

x→+∞

[a, +∞)

✈➔

lim [g(x) − A] = 0✳

x→+∞

❚ø ❛✮ s✉② r❛

b
r➡♥❣ ❤➔♠ sè

b→

f (x)dx ❜à ❝❤➦♥ tr➯♥ [a, +∞)✳ ❚❤❡♦ ❞➜✉ ❤✐➺✉ ❉✐r✐❝❤❧❡t✱
a
+∞

f (x)[g(x) − A]dx

tø ✤â t❛ s✉② r❛ t➼❝❤ ♣❤➙♥

a
✶✷

❤ë✐ tö✳


+∞

f (x)dx

❱➻

❤ë✐ tö ✈➔

a
+∞

+∞

f (x)[g(x) − A]dx +

f (x)g(x)dx =
a

+∞

a

Af (x)dx
a

+∞

f (x)g(x)dx

♥➯♥ t➼❝❤ ♣❤➙♥

❤ë✐ tö✳

a

✶✳✸ ◆❤➢❝ ❧↕✐ ♠ët sè ❦❤→✐ ♥✐➺♠ tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥
✣à♥❤ ♥❣❤➽❛ ✶✳✸✳
P❤÷ì♥❣ tr➻♥❤ ❞↕♥❣

dy
+ P (x)y = Q(x)
dx

✭✶✳✶✮

✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ♠ët✳

P (x)

❚r♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✮ t❛ ❧✉ỉ♥ ♠➦❝ ✤à♥❤
tr➯♥ ❦❤♦↔♥❣

(a, b)

✈➔

Q(x)

❧➔ ①→❝ ✤à♥❤

♥➔♦ ✤â✳ ◆❣❤✐➺♠ tê♥❣ q✉→t ừ ữỡ tr t

ữủ ữợ

y = e

✶✳✹✳

P (x)d(x)

[

Q(x)e

P (x)dx

dx + C].

P❤÷ì♥❣ tr➻♥❤ ❝â ❞↕♥❣

F (x, y, y , y , ..., y (n) ) = 0,
✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ❜➟❝

n✳

❚r♦♥❣ ✤â

✭✶✳✷✮

y = y(x)

❧➔ ❤➔♠

❝➛♥ ♣❤↔✐ t➻♠✳

✣à♥❤ ♥❣❤➽❛ ✶✳✺✳

❍➔♠

y = ϕ(x)

✤÷đ❝ ❣å✐ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

✈✐ ♣❤➙♥ t❤÷í♥❣ ✭✶✳✷✮ ♥➳✉ ♥❤÷ tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✮ t❤❛②

y = ϕ (x)✱✳✳✳✱ y (n) = ϕ(n) (x)

✣à♥❤ ♥❣❤➽❛ ✶✳✻✳

t❛ ♥❤➟♥ ✤÷đ❝✿

F (x, ϕ(x), ϕ (x), ..., ϕ(n) (x)) = 0.
P❤÷ì♥❣ tr➻♥❤ ❞↕♥❣

y + py + qy = f (x)
✶✸

y = ϕ(x)✱


ð ✤➙②

p, q

❧➔ ♥❤ú♥❣ ❤➡♥❣ sè✱ ❝á♥

f (x)

❧➔ ♠ët ❤➔♠ ✤➣ ❜✐➳t ✭✤÷đ❝ ♠➦t ✤à♥❤

❧➔ ①→❝ ✤à♥❤ ✈➔ ❧✐➯♥ tư❝ tr➯♥ ♠ët ❦❤♦↔♥❣ ♥➔♦ ✤â✮✱ ✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤
✈✐ ♣❤➙♥ ❝➜♣ ❤❛✐ ❤➺ sè ❤➡♥❣✳

✣à♥❤ ♥❣❤➽❛ ✶✳✼✳

P❤÷ì♥❣ tr➻♥❤ ❞↕♥❣

d(n−1) y
dy
dn y
A0 n + A1 (n−1) + ... + An−1 + An y = 0
dx
dx
dx
✈ỵ✐

Ai ✱ i = 0, n ❧➔ ❤➡♥❣ sè✱ A0 = 0✱ ✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥

t➼♥❤ ❝➜♣ ♥ t❤✉➛♥ ♥❤➜t ✈ỵ✐ ❤➺ số tữỡ ự ợ õ ữỡ tr

dy
dn y
d(n1) y
A0 n + A1 (n−1) + ... + An−1 + An y = f (x)
dx
dx
dx
✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ♥ ❦❤ỉ♥❣ t❤✉➛♥ ♥❤➜t✳

✣à♥❤ ♥❣❤➽❛ ✶✳✽✳

P❤÷ì♥❣ tr➻♥❤ ❞↕♥❣

y (n) + P1 (x)y (n−1) + ... + Pn−1 (x)y + Pn (x)y = f (x)
✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❜➟❝ ♥ ✈ỵ✐ ❤➺ sè ❜✐➳♥

Pn (x)

✭tr♦♥❣ ✤â

❦❤♦↔♥❣

(a, b)

P1 (x), P2 (x), ..., Pn (x)

♥➔♦ ✤â✮✳

✶✹

P1 (x), P2 (x), ...,

①→❝ ✤à♥❤ ✈➔ ❧✐➯♥ tö❝ tr♦♥❣ ♠ët


❈❤÷ì♥❣ ✷

P❍➆P ❇■➌◆ ✣✃■ ▲❆P▲❆❈❊

✷✳✶ P❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t
✣à♥❤ ♥❣❤➽❛ ✷✳✶✳ f (t)
✤ê✐ ▲❛♣❧❛❝❡

●✐↔ sû

❝õ❛ ❤➔♠

✤à♥❤ ♥❣❤➽❛

▲

❧➔ ♠ët ❤➔♠ sè ❧✐➯♥ tư❝ tø♥❣ ❦❤ó❝✱

f (t)

❧➔ t→❝ ✤ë♥❣ ❝õ❛ t♦→♥ tû

▲{f (t)} =

✈➔♦ ❤➔♠

♣❤➨♣ ❜✐➳♥
f (t)

✤÷đ❝

+∞

e−st f (t)dt = F (s),

✭✷✳✶✮

0
+∞

e−st f (t)

♥➳✉ t➼❝❤ ♣❤➙♥

❚♦→♥ tû

❱➼ ❞ö ✷✳✶✳
●✐↔✐✿

▲

❤ë✐ tö✳

0
t→❝ ✤ë♥❣ ✈➔♦ ❤➔♠ ❣è❝

f (t)

❝❤♦ t❛ ♠ët ❤➔♠ ↔♥❤

❚➻♠ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❝õ❛ ❤➔♠

❚ø ❝æ♥❣ t❤ù❝ ✭✷✳✶✮ t❛ ❝â✿

+∞

n

e−st dt = lim

F (s) =

=

✵ t❤➻✿

0

n
−st

e

e−st dt.

n→+∞

0
◆➳✉ s

f (t) = 1.

1
dt = − e−sn

s

0

✶✺

n
0

1 − e−sn
=
.
s

F (s)✳


❉♦ ✤â✿

n

lim

n→+∞
0


1
e−st dt = s
∞

s > 0,
s < 0.

◆➳✉ s ❂ ✵ t❤➻✿

n

n

e−st dt = lim

lim

n→+∞

1dt = lim n = +∞.

n→+∞

0

n→+∞

0

◆❤÷ ✈➟②✱ t➼❝❤ ♣❤➙♥ tr➯♥ ❝❤➾ ❤ë✐ tư ❦❤✐

s > 0✱

❦➳t q✉↔ ❧➔✿

+∞

e−st dt =

F (s) =

❱➼ ❞ö ✷✳✷✳
●✐↔✐✿

1
s

khi s > 0.

0
❚➻♠ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❝õ❛ ❤➔♠

❚ø ❝æ♥❣ t❤ù❝ ✭✷✳✶✮ t❛ ❝â✿

+∞

n

e−st tdt = lim

F (s) =

=

0

✵ t❤➻✿

+∞

e−st tdt = −
0
◆➳✉

e−st tdt

n→+∞

0
◆➳✉ s

f (t) = t.

s=0

t❤➻✿

+∞

−st +∞

te
s

+
0

1
1
e−st dt = −[ + 2 ]e−st
s s

1
s
0

+∞

t2
tdt =
2

0

+∞
0


 1 , s > 0,
=
s2
 ∞, s < 0.

+∞

== ∞.
0

◆❤÷ ✈➟②✱ t➼❝❤ ♣❤➙♥ tr➯♥ ❝❤➾ ❤ë✐ tö ❦❤✐ s ❃ ✵✱ ❦➳t q✉↔ ❧➔✿

L{t} = F (s) =

❱➼ ❞ö ✷✳✸✳

1
.
s2

❚➻♠ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❝õ❛ ❤➔♠

✶✻

f (t) = eat .


❚ø ❝æ♥❣ t❤ù❝ ✭✷✳✶✮ t❛ ❝â✿

+∞

e−st .eat dt

F (s) =
0

n

e−(s−a)t dt

= lim

n→+∞
0

e−(s−a)t n
| .
= lim
n→+∞ −(s − a) 0
◆➳✉

s−a>0

❱➼ ❞ö ✷✳✹✳
●✐↔✐✿

❚➻♠

❚❛ ❝â✿

e−(s−a)n → 0

n → +∞✱ ❞♦ ✤â t❛ ❝â✿
1
F (s) = −1 {eat } =

, s > a.
s−a
♣❤➨♣ ❜✐➳♥ ✤è✐ ▲❛♣❧❛❝❡ ❝õ❛ f (t) = sin at✱ g(t) = cos at✳

t❤➻

▲

▲

❦❤✐

+∞

e−st sin atdt✱

F (s) = {sin at} =
0

▲

+∞

e−st . cos atdt.

G(s) = {cos at} =
0
◆➳✉

s≥0

t❤➻✿

e−st
a
F (s) = −
sin at |+∞
+
0
s
s
a
⇒ F (s) = G(s);
s
a
e−st
G(s) = −
cos at |+∞
−
0
s
s

+∞

e−st cos atdt,
0

+∞

e−st sin atdt
0

⇒ G(s) =
❉♦ ✤â t❛ ❝â✿

1 a
− F (s).
s s

1 a2
− G(s)
s s2
s
⇒G(s) = 2
, s ≥ 0,
s + a2
a
⇒F (s) = 2
, s ≥ 0.
s + a2
G(s) =

✶✼


❱➼ ❞ö ✷✳✺✳
❚❛ ❝â✿

❚➻♠ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❝õ❛ ❤➔♠

e +e
2

f (t) = cosh at =

❉♦ ✤â✿

f (t) = cosh at.

−at

at

.

▲

▲ e2 + ▲{ e 2 }
at

−at

F (s) = {cosh at} = {

1 1
1 1
+ .
.
= .

2 s−a 2 s+a

▲{e } = s −1 a khis ≥ a,
▲{e } = s +1 a khis ≥ −a.
s
F (s) = ▲{cosh at} =
s ≥ |a|.
s −a
❇↔♥❣ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❝❤♦ ❝→❝ ❤➔♠ ✤➦❝ ❜✐➺t
❙❚❚ ❍➔♠ ❣è❝ ❢✭t✮ ❍➔♠ ↔♥❤
❋✭s✮
▼✐➲♥
❤ë✐
tö
1

❱➻✿

at

−at

◆➯♥✿

2

1

1

2

t

3

tn

4

eat

5

tn eat

6

cos at

7

sin at

8

cosh at

9

sinh at

10

eat cos kt

11

eat sin kt

2

❦❤✐

s>0

s
1
s2
n!

s>0

sn+1
1
s−a
n!
(s − a)n+1
s
s 2 + a2

a
s 2 + a2
s
s 2 − a2
a
s 2 − a2
s−a
(s − a)2 + k 2
k
(s − a)2 + k 2

s>0
s>a
s>a
s>0
s>0
s > |a|
s > |a|
s>a
s>a

❚➼♥❤ ❝❤➜t ✷✳✶✳ ▲{c f (t) + c f (t)} = c F (s) + c F (s).
1 1

2 2

1 1

✭❈❤ù♥❣ ♠✐♥❤ ❝â t❤➸ t❤❛♠ ❦❤↔♦ t➔✐ ❧✐➺✉ ❬✶❪✮

✶✽

2 2


❱➼ ❞ö ✷✳✻✳
●✐↔✐✿

❚➻♠ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❝õ❛ ❤➔♠

f (t) = 4t2 − 3 cos 2t + 5e−t

⑩♣ ❞ö♥❣ ♣❤➨♣ ờ ừ ữợ ữủ

ð tr➯♥ t❛ ❝â✿

▲{4t − 3 cos 2t + 5e } = 4 s2! − 3 s s+ 4 + 5 s +1 1 = s8 − s 3s+ 4 + s +5 1
❚➼♥❤ ❝❤➜t ✷✳✷✳ ▲{e f (t)} = F (s + a).
−t

2

3

2

3

2

−at

❈❤ù♥❣ ♠✐♥❤✿

▲{e

−at

❚❛ ❝â✿

+∞

+∞

[e−at f (t)]e−st dt =

f (t)} =
0

❚➼♥❤ ❝❤➜t ✷✳✸✳ ▲

f (t)e−(s+a)t dt = F (s + a).
0


0,
−as
{f (t−a)u(t−a)} = e F (s)✱ ✈ỵ✐ u(t−a) =
1,

❈❤ù♥❣ ♠✐♥❤✿

▲{f (t − a)u(t − a)} =

t < a,
t > a.

+∞

[f (t − a)u(t − a)]e−st dt
0
+∞

f (t − a)e−st dt

=

a
+∞

f (x)e−s(x+a) dx

=
0

+∞

= e−as

f (x)e−sx dx = e−as F (s).

0

❚➼♥❤ ❝❤➜t ✷✳✹✳ ▲{f (at)} = a1 F ( as ),
❈❤ù♥❣ ♠✐♥❤✿

❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡✱ t❛ ❝â✿

+∞

+∞
st

x

f (at)e− dt =

L{f (at)} =

a > 0.

f (x)e−s a

dx 1 s
= F ( )✱ a > 0.
a
a a

❚➼♥❤ ❝❤➜t ✷✳✺✳ ▲{f (t)} = sF (s) − f (0).
0

❈❤ù♥❣ ♠✐♥❤✿

0

❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ t❛ ❝â✿

+∞

+∞
− st

L{f (t)} =

f (t)e

dt = f (t)e−st |+∞
+s
0

0

f (t)e−st dt = sF (s) − f (0).
0

✶✾


❚÷ì♥❣ tü t❛ ❝â t➼♥❤ ❝❤➜t ✻ ✈➔ ✼✳

❚➼♥❤ ❝❤➜t ✷✳✻✳ ▲{f (t)} = s F (s) − sf (0) − f (0).

❱➼ ❞ö ✷✳✼✳
▲{sin at} = s +a a
2

❈❤ù♥❣ ♠✐♥❤✿

❚❛ ❝â✿

2

2

f (t) = sin at ⇒ F (s) = L{f (t)}✱

s✉② r❛✿

f (t) = a cos at, f (t) = −a2 sin at, f (0) = 0, f (0) = a
❚❤❡♦ t➼♥❤ ❝❤➜t ✻ t❛ ❝â✿

L{f (t)} = L{ − a2 sin at} = −a2 L{ sin at} = −a2 L{f (t)}
L{f (t)} = s2 F (s) − s.0 − a = s2 L{f (t)} − a
a
⇒ L{ sin at} = 2
.
s + a2

❚➼♥❤ ❝❤➜t ✷✳✼✳ ▲{f
❚➼♥❤ ❝❤➜t ✷✳✽✳ ▲{

(n)

(t)} = sn F (s)−sn−1 f (0)−sn−2 f (0)−...−f (n−1) (0).

t

f (u)du} =

F (s)
.
s

0
t

❈❤ù♥❣ ♠✐♥❤✿

✣➦t

f (u)du ⇒ g (t) = f (t), g(0) = 0✳

g(t) =

❚ø t➼♥❤

0
❝❤➜t ✺ t❛ ❝â✿

L{g (t)} = sG(s) − g(0) = sG(s) = L{f (t)} = F (s) ⇒ G(s) =
t

⇒ L{g(t)} = L{

f (u)du} =

F (s)
s

F (s)
.
s

0

❚➼♥❤ ❝❤➜t ✷✳✾✳ ▲

dn
{t f (t)} = (−1)
F (s) = (−1)n F (n) (s).
n
ds
n

n

❈❤ù♥❣ ♠✐♥❤✿
+∞

+∞
−st

f (t)e dt ⇒ F (s) =

F (s) =
0

f (t)

∂ −st
e dt = −L{tf (t)}.
∂s

0

❙✉② r❛✿

L{tf (t)} = −

d
F (s) = F (s).
ds
✷✵


ỵ

ỵ tr
f (0+) = lim sF (s).
s→∞

❈❤ù♥❣ ♠✐♥❤✿

❚ø ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❝õ❛ ✤↕♦ ❤➔♠ ✭❚➼♥❤ ❝❤➜t ✺✮✿
▲➜② ❣✐ỵ✐ ❤↕♥ ❦❤✐

s → +∞
lim

s→+∞

♠➔

lim

s→+∞

▲{f (t)} =

▲{f (t)} =

▲{f (t)} = sF (s)−f (0+).

lim [sF (s) − f (0+)]

s→+∞

+∞

f (t)e−st dt = 0.

lim

s→+∞
0

❱➟②
▼➔

lim [sF (s) − f (0+)] = 0.

s→+∞

f (0+)

❧➔ ❤➡♥❣ sè ♥➯♥

f (0+) = lim sF (s).

ỵ ỵ tr ố

s

F (+) = lim(sF (s)).
s→0

❈❤ù♥❣ ♠✐♥❤✿
❚ø ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❝õ❛ ✤↕♦ ❤➔♠ ✭❚➼♥❤ ❝❤➜t ✺✮✿
▲➜② ❣✐ỵ✐ ❤↕♥ ❦❤✐

s→0

▲

+∞

f (t)e−st dt = lim[sF (s) − f (0+)]

lim {f (t)} = lim

s→0

▲{f (t)} = sF (s)−f (0+).

s→0

s→0

0
+∞
♠➔

+∞

f (t)e−st dt = lim

lim

s→0

df (t) = f (+∞) − f (0+).

s→0

0

0

f (+∞) − f (0+) = lim[sF (s) − f (0+)].
s→0
❍❛② F (+∞) = lim(sF (s)).

❱➟②

s→0

✷✳✷ P❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ♥❣÷đ❝
✣à♥❤ ♥❣❤➽❛ ✷✳✷✳
▲{f (t)} = F (s) ⇒ f (t) = ▲

P❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ♥❣÷đ❝ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉✿

−1

✷✶

{F (s)}.

✭✷✳✷✮


✷✳ ❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❧❛♣❧❛❝❡ ♥❣÷đ❝
❚➼♥❤ ❝❤➜t ✷✳✶✵✳ ▲ {c F (s) + c F (s)} = c f (t) + c f (t).

❚➼♥❤ ❝❤➜t ✷✳✶✶✳ ▲ {F (s + a)} = e f (t) = e ▲ {F (s)}.
❚➼♥❤ ❝❤➜t ✷✳✶✷✳ ▲ {F (s)} = e ▲ {F (s − a)}.
❚➼♥❤ ❝❤➜t ✷✳✶✸✳ ▲ {F (s − a)} = e ▲ {F (s)}.
❱➼ ❞ö ✷✳✽✳
−1

1 1

2 2

−1

1 1

−at

−1

−1

−1

−at

−1

−at

2 2

at

−1

❳→❝ ✤à♥❤ ♣❤➨♣ ❜✐➸♥ ✤ê✐ ▲❛♣❧❛❝❡ ♥❣÷đ❝ ❝õ❛ ❤➔♠

F (s) =

1
.
s2 − 2s + 5

●✐↔✐✿
1
1
1
=
, F (s + 1) = 2
2
− 2s + 5 (s − 1) + 22
s + 22
1
1
⇒ L−1 {F (s + 1)} = sin 2t, L−1 {F (s)} = et sin 2t.
2
2
F (s) =

s2

❚➼♥❤ ❝❤➜t ✷✳✶✹✳ ▲ {e F (s)} = f (t − a)u(t − a).
❚➼♥❤ ❝❤➜t ✷✳✶✺✳ ▲ {F (as)} = a1 f ( a1 ), a > 0.

▲ {F (s)} = (−1) t ▲ {F (s)}
❚➼♥❤ ❝❤➜t ✷✳✶✻✳  ▲ {F (s)} = −1 ▲ {F (s)}
t
s+1
.
❱➼ ❞ö ✷✳✾✳
F (s) = ln
s−1
−1

−as

−1

−1

(n)

n n

n

−1

−1

−1

(n)

n

❚➻♠ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ♥❣÷đ❝ ❝õ❛ ❤➔♠

●✐↔✐✿

❚❛ ❝â✿

1
1
−
⇒ L−1 {F (s)} =e−t − et = −2sht.
s+1 s−
1
n
(−1)
−1 −1
2sht
−1
(n)
−1
L−1 {F (s)} =
L
{F
(s)}
⇒
L

{F
(s)}
=
L
{F
(s)}
=
.
tn
t
t
F (s) =

❚➼♥❤ ❝❤➜t ✷✳✶✼✳ ▲

+∞
−1

{

F (u)du} =

1
t



1

{F (s)}.

s

ú ỵ

r ờ ữủ sè ❝➛♥ ✤÷đ❝ ❝❤✉②➸♥ ✈➲

❞↕♥❣ ♥❤÷ ❜↔♥❣ s❛✉✿

✷✷


❉↕♥❣ ❝õ❛ ♠➝✉ sè

❈❤✉②➸♥ ✈➲ ❞↕♥❣ ❝õ❛ ♣❤➙♥ t❤ù❝

A
ax + b
A1
A2
Ak
+
+ ... +
2
ax + b (ax + b)
(ax + b)k
A2 x + B2
Ak x + Bk
A1 x + B1
+

+
...
+
ax2 + bx + c (ax2 + bx + c)2
(ax2 + bx + c)k

ax + b
(ax + b)k
(ax2 + bx + c)k

✷✳✸ ỵ t

P ❝❤➟♣ ✭t➼❝❤ ❝❤➟♣✮ ❝õ❛ ❤❛✐ ❤➔♠
❤✐➺✉ ❣✐ú❛ ❝❤ó♥❣ ❜ð✐ ❞➜✉

f g

f (t)



g(t)

tũ ỵ ữủ ỵ

s

t

[f g](t) =

f (t − τ )g(τ )dτ ;
0

❤♦➦❝ ❜✐➸✉ ❞✐➵♥ q✉❛ ♠ët t➼❝❤ ♣❤➙♥ ✈ỉ ❤↕♥ ✤è✐ ✈ỵ✐ ❤❛✐ ❤➔♠

+∞

f g =
f g



h



g

tũ ỵ

+

f ( )g(t )d =




f

g( )f (t )d.


tũ ỵ

a

❤➡♥❣ sè✱ t➼❝❤ ❝❤➟♣ ❝õ❛ ❤❛✐ ❤➔♠

sè ❝â t➼♥❤ ❝❤➜t✿

f ∗ g = g ∗ f ; f ∗ (g ∗ h) = (f ∗ g) ∗ h;
f ∗ (g + h) = (f ∗ g) + (f ∗ h); a(f ∗ g) = (af ) ∗ g = f ∗ (ag).
▲➜② ✤↕♦ ❤➔♠ t➼❝❤ ❝❤➟♣✱ t❛ ❝â✿ (f ∗ g) = f ∗ g = f ∗ g ✱ ❤♦➦❝ ❧➜②
♣❤➙♥ t❛ ❝â✿

+∞

+∞ +∞

(f ∗ g)dt =
−∞

[

f (u)g(t − u)du]dt

−∞ −∞
+∞

=

+∞

f (u)[
−∞
+∞

=[

g(t − u)dt]du

−∞
+∞

f (u)du][

−∞

−∞
✷✸

g(t)dt],

t➼❝❤


tr♦♥❣ ✤â t➼❝❤ ♣❤➙♥ ♥➔② ❦❤æ♥❣ t❤❛② ✤ê✐ ❣✐→ trà ❦❤✐ ✤ê✐ ❝❤é ❝→❝ ❤➔♠
✈➔

g(t)✱

f (t)

❞♦ ✤â t➼❝❤ ❝❤➟♣ ❝õ❛ ❤❛✐ ❤➔♠ ✤è✐ ①ù♥❣ ♥❤❛✉ ✤è✐ ✈ỵ✐ ❝→❝ ❤➔♠ ♥❤➙♥

❝❤➟♣✳

❱➼ ❞ö ✷✳✶✵✳

❚➼❝❤ ❝❤➟♣ ❝õ❛

cos t

sin t

✈➔

❧➔✿

t

(cos t) ∗ (sin t) =

cos τ sin(t − τ )dτ.
0

⑩♣ ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ ❧÷đ♥❣ ❣✐→❝

1

cos A sin B = [sin(A + B) − sin(A − B)].
2
❚❛ ❝â✿

t

1
[sin t − sin(2τ − t)]
2

(cos t) ∗ (sin t) =
0

1
1
= [τ sin t + cos(2τ − t)]tτ =0 ,
2
2
õ

ỵ

1
(cos t) (sin t) = t sin t.
2

f (t) ✈➔ g(t) ❧➔ ♥❤ú♥❣ ❤➔♠ ❧✐➯♥
ct
tö❝ ✈ỵ✐ t ≥ 0 ✈➔ |f (t)|✱ |g(t)| ✤➲✉ ❜à ❝❤➦♥ ❜ð✐ M e ✈ỵ✐ t → ∞✳ ❑❤✐ ✤â
ờ ừ t f g tỗ t↕✐ ❦❤✐ s > c✱ ❤ì♥ ♥ú❛✿

✭❳❡♠ t➔✐ ❧✐➺✉ ❬✼❪✮

●✐↔ sû

▲{f (t) ∗ g(t)} = ▲{f (t)}▲{g(t)} = F (s).G(s)
▲

✈➔

−1

{F (s).G(s)} = f (t) ∗ g(t).

◆❤÷ ✈➟② ❞ị♥❣ t➼❝❤ ❝❤➟♣ ❝❤ó♥❣ t❛ ❝â t❤➸ t➻♠ ✤÷đ❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❝õ❛

F (s).G(s)

♥❤÷ s❛✉✿

▲

t
−1

{F (s).G(s)} =

f (τ )g(t − τ )dτ.
0

✷✹

❱➼ ❞ö ✷✳✶✶✳ f (t) = sin 2t g(t) = e
▲ { (s − 1)(s2 + 4) } = sin(2t) ∗ e = e
❱ỵ✐

t

✈➔

✱ t❛ ❝â✿

t

−1

t

t−τ

2

sin 2τ dτ

0
t

e−τ sin 2τ dτ = et [

= et

❱➟②✿

▲

e−τ
(− sin 2τ − 2 cos 2τ )]t0 .
5

0
−1

{

2 t 1
2
2
}
=
e
−
sin
2t
−
cos 2t.
(s − 1)(s2 + 4)
5
5
5

✷✺