!"MIDTERM #$%& #$%& MATH 3 @facebook live
!"!"#$ !"#$ M ATHEMATICS 3 BY P’GOD
1 THE LAPLACE TRANSFORM LAPLACE TRANSFORMS
F(t)
f(s)
F(t)
f(s)
"
L{F(t)}
=
#st ! e F(t)dt
=
f(s)
e
1
at
s!a
0
1 1
s
s
1 t
s
t
s
sa
!"#$%&'(()
s
1)
1
2
+ b
2
2
! b 2 b
sinh bt
+
2
s
2
! b 2
: Gamma function "
#u
n
u du
,
1. "( a + 1) = a"(a )
,
=
+
+ b
s
cosh bt
n +1
!(a
a
%(n + 1)
2
2
b
sin bt
n!
n
s
t
s
cos bt
! e
n $1
0
3. !(1)
=
1
a >
!1
2. !( n + 1) = n!
1
4. " ( )
2
-1-
=
,
n =
0 ,1, 2 , …
!
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!"!"#$ !"#$ M ATHEMATICS 3 BY P’GOD Example1 1. F(t)
=
!"#$%&'$()*&"&$*&$+,-"(./010$2)
1
Example2
2. F(t)
=
e at
!"#$%&'$()*&"&$*&$+,-"(./034( )
4
3/2
1. L{t }
2. L{t
3. L{cos2t}
4. L{sinh 2 t}
-2-
}
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!"!"#$ !"#$ M ATHEMATICS 3 BY P’GOD
()%*+,-,)+,).... /012345!" !" *(+, 1.
*(+,-."-/%
2.
e -01%
3.
t 23
4.
5.
F(t)
f(s) = L{F(t)}
aF(t) + bG(t)
af(s) + bg(s)
e at F(t)
f(s ! a)
n
t F(t)
(!1)
F (n) (t)
s n L{F(t)} ! s n !1 F(0) ! ... ! F (n !1) (0)
! F(! )d!
f(s)
s
s
0
7.
8.
!"#$%=)>
F(t + p)
!"#$%?%+%@A
[f(s)]
t
167%8
9%:';<
ds
n
! F(u)du
t
6.
dn
"
F(t)
t 4)5
n
=
1
F(t)
1" e
p
" st e F(t)dt " sp ! 0
e! as L{F(t)}
F(t ! a)u(t ! a)
=
e! asf(s)
F(t) * G(t) 9.
Convolution
t
L{F(t)} ! L{G(t)}
! F( # )G(t " # )d #
=
f(s) ! g(s)
=
0
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!"!"#$ !"#$ M ATHEMATICS 3 BY P’GOD
!5#617 #617))-... 2345!" !"8&8 1 -(:;5 ;5<=;" <=;"-=2> -=2>< & 7 1 : ()%*+,-,)+,).96-$5-(: 1. !"#$5$ L{cos 2t} 2
&2 3 # 't ' sin4t " + t + 2e $3 !
2. !"#$5$ L%
3. !"#$5$ L{t
4
+
cos3t ! 3e !2t
+
sinh4t}
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!"!"#$ !"#$ M ATHEMATICS 3 BY P’GOD
!5#617 #617))-... 2345!" !"8&8 2 -(:;5 ;5<8& <8& ?@? 7 A0> A @ 0>#1 #1 eat & 7 2 : ()%*+,-,)+,).96-$5-(: 1. !"#$5$6-7*8 1. L{4te ! t }
2.
2. L{e 4t t 3 }
!"#$%&'$()*&"&$*&$+,-" F(t)
=
3. L{e !2t cosh3t}
e ! t (3sinh2t ! 5cosh2t)
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!"!"#$ !"#$ M ATHEMATICS 3 BY P’GOD
!5#617 #617))-... 2345!" !"8&8 3 -(:;5 ;5<8& <8& ?@? 7 A0> A @ 0>#1 #1 & 7 3 : ()%*+,-,)+,).96-$5-(: 1. !"#$5$,-" L{t e 2
!2t
t
}
2. !"#$%&'$()*&"&$*&$+,-" F(t)
=
te 2t cos3t
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!"!"#$ !"#$ M ATHEMATICS 3 BY P’GOD
!5#617 #617))-... 2345!" !"8&8 4 -(:;5 ;5<8& <8& B)%0> B 7 )%0>#1 #1 & 7 4 : ()%*+,-,)+,).96-$5-(: 1. !"#$%&'$()*&"&$*&$+,-" F(t)
t
sin2t
=
t
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!"!"#$ !"#$ M ATHEMATICS 3 BY P’GOD
!5#617 #617))-...2345!" !8&8" 6 -(:;5 ;5<6" <6"<8" <8"(%5 (%5, & 7 6 : ()%*+,-,)+,).96-$5-(: t
1. !"#$%&'$()*&"&$*&$+,-" F(t)
=
! sinh d !
!
!
0
-8-
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!"!"#$ !"#$ M ATHEMATICS 3 BY P’GOD
!5#617 #617))-... 2345!" !"8&8 7 -(:;5 ;5
#e $ t f(t) = " !1
,
0
, 1< t
< <
1 2
)&9
f(t + 2) = f(t)
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!"!"#$ !"#$ M ATHEMATICS 3 BY P’GOD 3. :#;/ f(t)
<"=* !"#$
L{f(t)}
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!"!"#$ !"#$ M ATHEMATICS 3 BY P’GOD
!5#617 #617))-..2345!" !8&8" 8 -(:;5 ;5<95 <95 <45 <> 45 45"?@;
u a (t) u a (t)
=
#A-
u(t ! a) B"10$2./0
u(t & a)
=
#0 , 0 % t % a " t$a !1 ,
#C''$(DE0;>"?@; f(t) FG-HF;=*,-">"?@;I;J;7/#K"#LM0 !9:#;/ a !$'5$NO/,-" t F;)6&9PM" DP;
f(t)
=
$ f 1 (t) , 0 & t < 2 ! #f 2 (t) , 2 & t < 4 !f (t) , t%4 "3
f(t) = c 0 u 0 (t) + c1 u 2 (t) + c 2 u 4 (t)
./0Q
c0 c1 c2
Example
1. f(t)
=
!"DE0;>"?@;
f(t)
u a (t)
=
=
=
f 1 (t) f 2 (t) ! f 1 (t) f 3 (t) ! f 2 (t)
Q:#;/FG-HF;=*%&RM'DS"DT;,-">"?@;I;J;7/#K"#LM0
#sint , 0 % t % ! " t$! ! t ,
2.
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$ t !'1 ! f(t) = # !t ' 4 !"e 5' t
,
0&t
<
2
,
2&t
<
4
,
4&t
<
5
,
t%5
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!"!"#$ !"#$ M ATHEMATICS 3 BY P’GOD
-(:;5 ;5<95 <95 <45 <> 45"?@; " 3/2 !t
+
t 3/2
,
tsin(t % 3)
, 3< t
0
< <
3
$
1. !"DE0; f(t) F;=*%&(M2DS"DT;,-">"?@;I;J;7/#K"#LM0 2. !"#$ L{f(t)}
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!"!"#$ !"#$ M ATHEMATICS 3 BY P’GOD
!5#617 #617))-... 2345!" !"8&8 9 H,()%+%I2)<) & 7 9 : Convolution (8EFG&H,()%+%I2)< &t # ' ! ! L e sin(t )d 1. !"#$5$ % ( " $0 ! !
& t 2 '3 # )d! " 2. !"#$5$ L % ) ! e ( cos4(t ' ! )d $0 ! !
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!"!"#$ !"#$ M ATHEMATICS 3 BY P’GOD
2 THE INVERSE LAPLACE TRANSFORM BC)( L{F(t)}
L!1{f(s)}
=
=
f(s)
F(t)
DE')54)F<')5&G<"<)G<)HF'I% 1. '$(FU4$($"'$()*&"&$*&$+ (2-")RRV-;'CR) 2. '$()0'DW;DXYZM;[-0 : Partial Fraction '$()0'DXYZM;[-0 : Partial Fraction 1.
2.
P(s) (s ! a)(s ! b) … (s ! n) P(s) (s ! a) n
3.
A
A
=
(s ! a)
P(s) as 2 + bs + c
+
=
s
B
!a
B (s ! b) 2
+
+
s
…+
N
! b
+
…+
s!n
N (s ! n) n
As + B
=
as
2
+ bs +
c
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!"!"#$ !"#$ M ATHEMATICS 3 BY P’GOD
!5#617 #617))-... 2345!" !" & 3s + 5 # % " s 7 + $ !
1. !"#$5$
L
2. !"#$5$
L
3. !"#$5$
L'1 %
4. !"#$5$
L
'1
'1
2
&s+2 3 # ' " % $s + 1 s ! 2
&
2
1
$ (s ' 1)
'1
& % $s
3
+
# " s 2 + 2s ' 8 ! 1
# " + 6s + 11! s
2
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' $ . + 3 s 2 7 s 2 ! ! * .1 + ln + 5. !"#$5$ L & 5 . (# 3 s+ 2 , s + 1 )! !% s 2 "
6. !"#$5$ L
& ' %tan $
7. !"#$5$ L
& s +1 # % " $s ' s !
'1
'1
1
3
!"!"#$ !"#$ M ATHEMATICS 3 BY P’GOD
1#
"
s!
2
- 16 -
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!"!"#$ !"#$ M ATHEMATICS 3 BY P’GOD 8. !"#$
'1
L
&4 e' % ' $s s
s
2
e +
s
2
'3s
# " ' 9!
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!"!"#$ !"#$ M ATHEMATICS 3 BY P’GOD
3 APPLICATIONS OF THE LAPLACE TRANSFORM ')54)JK1>L1"*(')5-."167%8 1. FZ&$*&$+ Eample1
"
2. \/=*
3. FZ&$*&$+%'];
"
!")^_`#$5$Da2b; y !! + 6 y ! + 34 y
=
0
,
y (0)
=
3 ,
- 18 -
y !(0)
=
1
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!"!"#$ !"#$ M ATHEMATICS 3 BY P’GOD
/J81:... (!:,)+,). ,)+,). ... ()%+%I1K(!: 1. !")^_`#$5$Da2b; y !! " 2 y ! + 5 y
= "
8e
" t
, y (0) = 2 , y !(0) = 12
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