ページ3:

3- shift #
5-space
的位移
-(s-a)
=
L [ eat fl] = So est eat fit) dt Soethar fit) dt F(S-a)
at
可先處理e以外的部分,再位移
EX. Initial value problem
y" - y = t
,
9'10)=1
y (0) -1
=
£ [y" -y] L [t] +
L[Y] = \\s)
D 1
L [y'] = Soesty it) de
∞
=
· (e" Jo) - <s√ ye" de
(-s) fo est
。
(0-y (o) + Sĭ (5)
L [y'] = - Y (0) + Sĭ (S)
L [y"] = -y' (o) + SY'(S)
-y' (0) + 5 (-y (0) + ST(S))
"L[y"] = y' 10) - syio) +5²Y (5)
b
I
Ⓒ est
-St
·-se'

ページ4:

Je t⋅ space / 2X 5-space
(y` 10) - sy (+) +5° | (5)) -(((5)) +
A
解Y(5) 題目:y/10)=1,y (0) = 1
=
( 1 - 5 + s ² Y ( 5 )). Y(S) (2
(5²-1) ĭ (5)
+
Y (5) = 5 + + (5-1).
(531)
FA JE S-space
S+.1
+
53(531)
fox t-space
t-space y (t) = ?
Y(S)
5-1
查表
|查
| (+)
y\t).
=
=
[+] + [+] + £[]
-t + sinht + et

ページ5:

Hearside function 單位階躍函數
u (t-a) [
1:
=
0
t>a
t-o
L [ult_a)]
ult-a).
a
N
So est ult-a) dt
-St
- So' est ult. a) dt + Sa e stult-a) dt
8
Sa est lat
e
-st
-sa
(-5)
0-
Le² + esa
˙的位移
t-shift
>
t-space
aso, move to right with
=
a unit
otca 前面都被移除
f(t-a), t>a
fit) f (t-a)n (ta) { {
=
=
1 [ FH)] - So est fit a) u(t-a) It
·Set t-a-t
·t = c + a
0
-st
•S• et filter nito) de S. e* f(ta). 1 dz
=
Sa
=
flt-a) ult-a) dt +
So et fic dz
e-sa fó
-57
e flo) dz
a
ex I
7=0
先解:F(S).
12
esa £ [f(z)]. esa F(5) #4 1 13
=

ページ6:

EX. ecology. An endangered species;
ult-a).
a
原本
dy
released
"y(t)
alt-1)
ult-2)
-a
u (t-1)⋅ u (t->) -r(t)
It-ay
現在: dy
十
at ay + r(t)
=
y' + ay = 0
y' + ay - R. (·ult-T) - ult-T>))
=
L[y' + ay ] - L [R (ultT) -ult-T»))]
=
-ST
every
month Ti to Tz
t-unit
population 110)-y.
grouch rate
R releasing units.
R( sest test.)
-572
(sta) Y (5) - Yo Rest kesto
SY(S)-y 10) + aĭ (s)
=
=
R
1715) =
Yo t
-ST₁
-57₂
t-space space
St a
eat
S-a
y (t)
=
=
'L' [Y(s)]
Yo
Sta
+
|-
-ST,
Sta
-
-ST₂
· Yo eat + RL" (e" \(s)\sta)] - RL" [es (+) (sta))
Yo eat + RL" [e" \\ \)] - RL" [e (+)

ページ7:

Yo
-at k
-1 -571
J₁t) = J. e² + (L" (e" (+)) (e" (+))
=
a
-
yo eat + R (ult-T.) (1- e alt-T.)) -u (t-To) (1-e a (t-T))
t
a
·Yo eat, t-Ti
R (1- e alt-T₁))
,
Yo
'yo eat + &
T₁ etc T₂
Yo eat + R (-e
alt-T)
a
+ e alt-To)).
t > T ₂
·y(t),
Yo
T₁
Tz

ページ8:

EX y + zy' +zy = u(t-1)-u(t²),
"
ylo) = 0, y' (o) -
L [g" + 3y²+ >y] L [u (t-1)] - £ [u (t-2)]
=
S³Y(S) - 1 10) sy² (0) + 35ĭ (5)-3(10) + 2) (5)
= 0
Y(s)
èsè
-25
5(5²+35+2)
y (t) = L" [e³ F(s) ] = £" [e² F(s)]
F(S) =
·S (S+1) (S+²)
|
a
b
2
+
Stl
5+2
S
se se
a (S+1) (sta) + b (s) (S+3) +c (S) (8+1)=1
·when s=0; a=
5=-2 C ===
i
L^[F(S)]
=
-
-2ť
+
e
5+2
y (t) = L" [ e³ F(s) ] - £" [e's F(s)]
-u (t-1) f(t-1) -u (t-2) f(t-2)
-(t-1).
-(x-2)-2(x-2).
- ulei) (s - e(t + ")-ult) (±-et te "")
=
tel.
-(4-1)
·e + = e
->(t-1)
1<t<2
-(t-1)
-2(t-1)
-(t-²) -2(t-2)
2<t

ページ9:

Darc's delta function
t+a
§ it. a) (∞ t-a
970
£ [ 8 (ta)] = so est f (ta) dt S. est du(ta) dt
。
otto
Julta) =
=
dt.
∞ t=0
=
- f (t-a).
dt
hard to understand
a
another
微分的定義
u (t-a-1))
way (t-a) = lim + (u(ta) - ù (t-a
0
Qa+k
area =!
有限的寬度k
··\t-a) = lim fk
ko
e-as (1-ets)
I [f(ta)] = lim ks
-as
-
fk
L [fr] = ± (u(ta) -u (t-a-k))
= 1 (east - e (a+b)s + )
as 11-eks
ks
-(a+k)s
Un fet (Nets & elarb)s, eas

ページ10:

EX ecology: An endangered species released Rat T.
910=?
y.
y(t) population,ylo) - Yo
=
-a.
y' + ay - rit) = RS (t-T₁)
=
L[y' + ay] = L [RS (t. 7.)]
·SY (S)-y (0) + aĭ (s) = Retis
growth rate
R releasing units
Y(S) = Yo
-TS
Re
+
Sta
Sta
t-shift
-Tis
e
Sta
y(t) = L" [Y(s) ] = L" [ \ 1 ] + RL" [ { " ]
Sta
• Y. eat + R (u (t-T.) e alt-T.))
Yo eat
J. ea +R
at
it ≤ T₁
t‹ti
; t = T₁
at
· Yo e + Rea
alt-T₁)
t > T₁

ページ11:

EX.
·y" + 3y² +zy = f (t-1) y 10) = 0, y 101 = 0
· L [y " + 3y² + zy] = £ [ 8 (t-1)]
0
S³Y (5) -sy(0)-y (0) +3 5/15) -y (0) + 2(15) = es
Y(S) =
és
52+35+2
es (st² + 5+1)
St.2
y (t) = L' [ es (st 2 + 5+1)]
=
a (+-) (-e) e (t))
u
0
+
t"
• e-(t.₁) t>1
-2 (t:1)
e
+e