�(t) + 5Ź(t) + 6y(t) = 0
y(0+) = a, Ź(0+) = b.
s2Y (s)-sy(0+)-Ź(0+)+5(sY (s)-y(0+))+6Y (s) = 0
5a + b + as 5a + b + as
Y (s) = = .
6 + 5s + s2 (2 + s)(3 + s)
Y (s)
Y (s)
3a + b 2a + b
Y (s) = - .
2 + s 3 + s
y(t) = L-1(Y (s)) = (3a+b)�e-2t-(2a+b)�e-3t, t e" 0.
L(s) L(s)
y(t) = � e-2t + � e-3t, t e" 0
M (s) s=-2 M (s) s=-3
L(s) dM(s)
Y (s) = , M (s) = .
M(s) ds
M (s) = 5+2s
t
Ź(t) + 4 e(t-�)y(�)d� + 3y(t) = 1, y(0+) = 1.
0
sY (s) - y(0+) + 4 � L(et " y(t)) + 3Y (s) = 1/s.
s - 1 1 1
Y (s) = = -
s(1 + s) 1 + s s(1 + s)
y(t) = e-t - (1 - e-t) = 2e-t - 1, t e" 0.
y(3)(t) + 4y(2)(t) + 4y(1)(t) + 3y(t) = u(t)
u(t)
y(0+) = 1, y(1)(0+) = 2, y(2)(0+) = 3.
L(y(1)(t)) = sY (s) - y(0+)
L(y(2)(t)) = s2Y (s) - sy(0+) - y(1)(0+)
L(y(3)(t)) = s3Y (s) - s2y(0+) - sy(1)(0+) - y(2)(0+).
1 + 15s + 6s2 + s3
Y (s) = .
s(3 + 4s + 4s2 + s3)
1 0.80952 -4.28571 + 0.14286s
Y (s) = + - .
3s 3 + s 0.866032 + (0.5 + s)2
y(t) = 0.33333 + 0.80952e-3t
+ 5.03322e-5t � sin(0.86603t - 0.02839), t e" 0.
f(t)
1 + 3s
F (s) = .
1 + s + s2
g(t) = fŁ(t)
G(s) = L(g(t)) = sF (s) - f(0+).
s + 3s2
f(0+) = lim f(t) = lim sF (s) = lim = 3.
s" s"
t0+ 1 + s + s2
s + 3s2 -3 - 2s
G(s) = - 3 = .
1 + s + s2 1 + s + s2
-3s - 2s2
g(0+) = lim g(t) = lim sG(s) = lim = -2.
s" s"
t0+ 1 + s + s2
g(0+) f(t)
F (s)
1 + s + s2 = 3/4 + (1/2 + s)2.
t e" 0
" " "
f(t) = -1/ 3 � e-t/2 � sin( 3t/2) + 3e-t/2 � cos( 3t/2)
f(0+) = 3.
 f(t) t e" 0
" "
g(t) = - 3e-t/2 � sin( 3t/2)
"
- 2e-t/2 � cos( 3t/2).
g(0+) = -2.
�(t) + 3Ź(t) + 2y(t) = 0, y(0+) = a, Ź(0+) = b.
�(t) + 2Ź(t) + 5y(t) = 3 � 1(t), y(0+) = 0, Ź(0+) = 0.
�(t) + 4Ź(t) + 3y(t) = u(t)
u(t) = 2 cos 3t t e" 0 y(t)
 y(t) u(t) = E � 1(t)
C +Q0
ńł
Ź1
�ł - 2Ź2 + y1 = 1(t)
y1 + Ź2 - 2y2 = e-t � 1(t)
ół
y1(0+) = 1, y2(0+) = 0.
"
f(t) = 1(t - n), t e" 0.
n=1
n = 0, 2, 4, . . .
1 dla n d" t < n + 1
f(t) =
-1 dla n + 1 d" t < n + 2.
gn(t) = L-1(Gn(s))
n
Gn(s) = (i + s)-1, n " N.
i=1
n-1 n
Gn(s) = (2i + s) � (2i - 1 + s)-1, n " N.
i=1 i=1
"
F (s) = L(f(t)) = f(t)�e-stdt
0
x+j"
1
f(t) = L-1(F (s)) = F (s) � estds
2Ąj x-j"
L(ąf(t) + �g(t)) = ąF (s) + �G(s)
1 s
L(f(ąt)) = F (ą), ą > 0
ą
L(f(t - ą)) = F (s) � e-ąs
1
0
L(f(t) � es t) = F (s - s0)s
2
L(f (t)) = sF (s) - f(0+)
n
1(t) - 1(t - ą) L(f(n)(t)) = snF (s)- sn-if(i-1)(0+)
i=1
t
1
e-ąt L( f(�)d�) = F (s)
0 s
t �n �2
1
e-ąt L( � � � f(�1)d�1) = F (s)
0 0 0 sn
lim f(t) = lim sF (s)
t0+ s"
lim f(t) = lim sF (s)
t" s0
L(f(t) g(t)) = F (s) � G(s)
L-1(sF (s)G(s)) = f(t)g(0+) + f(t) " g (t)
F (s) = -L(tf(t))
(n)
F (s) = (-1)nL(tnf(t))
x=�+j"
1
L(f(t)g(t)) = F (x)G(s - x)dx
2Ąj x=�-j"
f(t) F (s) = L(f(t))
�(t) 1
1
1(t)
s
1
t2
s2
tn-1 1
(n-1)! sn
1
1(t) - 1(t - ą) (1 - e-ąs)
s
1
e-ąt
s+ą
tn-1 1
e-ąt, n > 0
(n-1)! (s+ą)n
1 1
(1 - e-ąt)
ą s(s+ą)
1 1
(e-ąt - e-�t)
�-ą (s+ą)(s+�)
�
1 ą 1
(1 + e-ąt - e-�t)
ą� ą-� ą-� s(s+ą)(s+�)
�
sin �t
s2+�2
s
cos �t
s2+�2
1 1
sin �t � e-ąt
� (s+ą)2+�2
1
n
"1 sin �n 1 - ś2t � e-ś� t
2
s2+2ś�ns+�n
�n 1-ś2
s+ą
cos �t � e-ąt
(s+ą)2+�2
1 1 � �
"
+ sin(�t - �) � e-ąt, � = - arctan
ą2+�2 � ą2+�2
ą s((s+ą)2+�2)
1 1
n
"1 sin(�n 1 - ś2t + �) � e-ś� t, � = arccos � s(s2+2ś�ns+�n)
-
2 2
�n �n 1-ś2
2
1 1
(ąt - 1 + e-ąt)
ą2 s2(s+ą)
1 1
(1 - e-ąt - ąt � e-ąt)
ą2 s(s+ą)2
N(s) n
1
F (s) = = Ak s-pk , pi = pj "i = j,

k=1
D(s)
n
k
f(t) = Ak�ep t,
k=1
N(s)
Ak = , k = 1, . . . , n
D (s)
s=pk
N(s) n
1
F (s) = = Ak s-pk , pi = pj "i = j,

k=0
sD(s)
n
k
f(t) = Ak � ep t, p0 = 0,
k=0
N(0) N(s)
A0 = , Ak = , k = 1, . . . , n
D(0) sD (s)
s=pk
N(s) n ni
1
F (s) = = Aij (s-pi)j ,
i=1 j=1
D(s)
pi - ni - i = 1, . . . , n,
n ni Aij
i
f(t) = �tj-1 � ep t,
i=1 j=1
(j-1)!
1 dni-j N(s)
Aij = � , j = 1, . . . , ni
(ni-j)! dsni-j s=pi
Di(s)
D(s)
Di(s) = , i = 1, . . . , n.
(s-pi)ni