Upload
others
View
10
Download
0
Embed Size (px)
Citation preview
EEngineering Mathematics Ingineering Mathematics I
Prof. Dr. Yong-Su Na g(32-206, [email protected], Tel. 880-7204)
Text book: Erwin Kreyszig, Advanced Engineering Mathematics,
9th Edition, Wiley (2006)
Ch 6 Laplace Ch 6 Laplace TransfomsTransfomsCh. 6 Laplace Ch. 6 Laplace TransfomsTransfoms6.1 Laplace Transform. Inverse Transform. Linearity.
s-Shifting
6.2 Transforms of Derivatives and Integrals. ODEsg
6.3 Unit Step Function. t-Shifting
6 4 Short Impulses Dirac’s Delta Function Partial Fractions6.4 Short Impulses. Dirac s Delta Function. Partial Fractions
6.5 Convolution. Integral Equations
6.6 Differentiation and Integration of Transforms
6.7 Systems of ODEs
6.8 Laplace Transform: General Formulas
6.9 Table of Laplace Transforms6 ab o ap a a o
2
Ch. 6 Laplace TransformsCh. 6 Laplace TransformsC 6 ap ace a s o sC 6 ap ace a s o s((라플라스라플라스 변환변환))
제어 시스템 해석, 설계 등 미분방정식을 풀 수 있는 도구로 공학전반에 널리 이용
Laplace Transform
상미분방정식을 라플라 변환하여 방정식 변환• 상미분방정식을 라플라스 변환하여 보조방정식으로 변환
• 대수적인 연산을 통하여 보조방정식을 푼다.
• 보조방정식의 해를 역변환하여 상미분방정식의 해를 구한다.
장점
• 비제차 상미분방정식의 해를 구할 때, 제차 상미분방정식의 일반해를따로 구할 필요가 없다.따 구할 필 가 없다
• 초기값은 보조방정식을 만드는 과정 중에서 자동적으로 고려된다.
• 불연속성, 순간적인 충격량, 또는 복잡한 주기함수를 입력으로 갖는 상미분방정식도쉽게 해를 찾을 수 있다쉽게 해를 찾을 수 있다.
Ch. 6 Laplace TransformsCh. 6 Laplace TransformsC 6 ap ace a s o sC 6 ap ace a s o s((라플라스라플라스 변환변환))
< Time domain과 s-Domain과의 관계>
66.1 .1 LapalceLapalce Transform. Inverse Transform.Transform. Inverse Transform.Shifti Shifti ss--Shifting Shifting
((라플라스라플라스 변환변환. . 역변환역변환. . 선형성선형성 그리고그리고 ss--이동이동))
Laplace Transform (라플라스 변환): ( ) ( ) ( )dttfefsF st∫∞
−==0
L
Inverse Transform (역변환):
0
( ) ( )tfF =−1L
Ex. 1 Let when . Find .0≥t( ) 1=tf ( )sF
( ) ( ) ( )0111 >====∞
−∞
−∫ sedtef ststLL( ) ( ) ( )0 100
>=−=== ∫ ss
es
dtef LL
Ex. 2 Let when . Find .0≥t( ) atetf = ( )fL
( ) ( ) 11
00 ase
sadteee tasatstat
−=
−==
∞−−
∞−∫L
66.1 .1 LapalceLapalce Transform. Inverse Transform.Transform. Inverse Transform.Shifti Shifti ss--Shifting Shifting
((라플라스라플라스 변환변환. . 역변환역변환. . 선형성선형성 그리고그리고 ss--이동이동))
Linearity of the Laplace Transform: 라플라스 변환은 선형연산이다.
( ) ( )( ) ( )( ) ( )( )tbtftbtf LLL ( ) ( )( ) ( )( ) ( )( )tgbtfatbgtaf LLL +=+
Ex.3 Find the transforms of andatcosh atsinh
( ) ( )atatatat eeateeat −− −=+=21sinh ,
21cosh
( ) ( ) 11( ) ( )as
e,as
e atat
+=
−= − 1 1
LL
66.1 .1 LapalceLapalce Transform. Inverse Transform.Transform. Inverse Transform.Shifti Shifti ss--Shifting Shifting
((라플라스라플라스 변환변환. . 역변환역변환. . 선형성선형성 그리고그리고 ss--이동이동))
Linearity of the Laplace Transform: 라플라스 변환은 선형연산이다.
( ) ( )( ) ( )( ) ( )( )tbtftbtf LLL ( ) ( )( ) ( )( ) ( )( )tgbtfatbgtaf LLL +=+
Ex.3 Find the transforms of andatcosh atsinh
( ) ( )atatatat eeateeat −− −=+=21sinh ,
21cosh
( ) ( ) 11( ) ( )as
e,as
e atat
+=
−= − 1 1
LL
( ) ( ) ( )[ ] 1111cosh seeat atat =⎟⎞
⎜⎛ +=+=⇒ −LLL( ) ( ) ( )[ ]
( ) ( ) ( )[ ] 22
22
1121
21sinh
22cosh
aeeat
asasaseeat
atat =⎟⎠⎞
⎜⎝⎛ −=−=
−⎟⎠
⎜⎝ +
+−
+⇒
−LLL
LLL
( ) ( ) ( )[ ] 2222 asasas −⎟⎠
⎜⎝ +−
66.1 .1 LapalceLapalce Transform. Inverse Transform.Transform. Inverse Transform.Shifti Shifti ss--Shifting Shifting
((라플라스라플라스 변환변환. . 역변환역변환. . 선형성선형성 그리고그리고 ss--이동이동))
66.1 .1 LapalceLapalce Transform. Inverse Transform.Transform. Inverse Transform.Shifti Shifti ss--Shifting Shifting
((라플라스라플라스 변환변환. . 역변환역변환. . 선형성선형성 그리고그리고 ss--이동이동))
First Shifting Theorem (제 1이동정리), s-Shifting (s-이동)
( )( ) ( ) ( )( ) ( ) ( ) ( ){ }asFtfeasFtfesFtf -atat −=−=⇒= 1 , LLL ( )( ) ( ) ( )( ) ( ) ( ) ( ){ }fff ,
Ex. 5 ( ) ( )( ) 2222 cos cos
ωω
ωω
+−−
=⇒+
=as
astes
st atLL
Use these formulas to find the inverse of the transform
( )
( ) ( )( ) 2222 sin sin
ωωω
ωωω
+−=⇒
+=
aste
st atLL
Use these formulas to find the inverse of the transform
( )4012
13732 ++
−=
sssfL
( )( ) ( ) ( )
( )ttess
ss
sf t 20sin720cos34001
2074001
13400114013
21
21
21 −=⎟⎟
⎠
⎞⎜⎜⎝
⎛
++−⎟⎟
⎠
⎞⎜⎜⎝
⎛
+++
=⎟⎟⎠
⎞⎜⎜⎝
⎛
++−+
= −−−− LLL
66.1 .1 LapalceLapalce Transform. Inverse Transform.Transform. Inverse Transform.Shifti Shifti ss--Shifting Shifting
((라플라스라플라스 변환변환. . 역변환역변환. . 선형성선형성 그리고그리고 ss--이동이동))
Existence Theorem for Laplace Transforms (라플라스 변환의 존재정리)
함수 가 영역 상의 모든 유한구간에서 구분적 연속인 함수.
( )( )tf 0≥t
어떤 상수 와 에 대해 (너무 빠른 속도로 값이 커지지 않음)
모든 에 대해 의 라플라스 변환 가 존재
( ) ktMetf ≤k M⇒ ks > ( )tf ( )fL
66.1 .1 LapalceLapalce Transform. Inverse Transform.Transform. Inverse Transform.Shifti Shifti ss--Shifting Shifting
((라플라스라플라스 변환변환. . 역변환역변환. . 선형성선형성 그리고그리고 ss--이동이동))
PROBLEM SET 6.1
HW 23HW: 23
66.2 Transforms of Derivatives and Integrals..2 Transforms of Derivatives and Integrals.ggODEs ODEs ((도함수와도함수와 적분의적분의 변환변환. . 상미분방정식상미분방정식))
Laplace Transform of Derivatives (도함수의 라플라스 변환):
( ) ( ) ( ) ( ) ( ) ( ) ( )0'0''0' 2 fsffsfffsf −−=−= LLLL( ) ( ) ( ) ( ) ( ) ( ) ( )000 fsffsf, ffsf LLLL
( )( ) ( ) ( ) ( ) ( ) ( )00'0 121 −−− −−−−= nnnnn ffsfsfsf LLProve!
Ex. 1 Let . Find ( ) tttf ωsin= ( )fL
66.2 Transforms of Derivatives and Integrals..2 Transforms of Derivatives and Integrals.ggODEs ODEs ((도함수와도함수와 적분의적분의 변환변환. . 상미분방정식상미분방정식))
Laplace Transform of Derivatives (도함수의 라플라스 변환):
( ) ( ) ( ) ( ) ( ) ( ) ( )0'0''0' 2 fsffsfffsf −−=−= LLLL( ) ( ) ( ) ( ) ( ) ( ) ( )000 fsffsf, ffsf LLLL
( )( ) ( ) ( ) ( ) ( ) ( )00'0 121 −−− −−−−= nnnnn ffsfsfsf LLProve!
Ex. 1 Let . Find ( ) tttf ωsin= ( )fL
( ) ( ) ( ) ( ) ttttffttttff ωωωωωωω sincos2'' ,00' ,cossin' ,00 2−==+==
( ) ( ) ( ) ( ) ( )22 2ωss( ) ( ) ( ) ( ) ( ) ( )22222
222sin 2'' ω
ωωωω
ω+
==⇒=−+
=⇒s
sttffsfs
sf LLLLL
66.2 Transforms of Derivatives and Integrals..2 Transforms of Derivatives and Integrals.ggODEs ODEs ((도함수와도함수와 적분의적분의 변환변환. . 상미분방정식상미분방정식))
Laplace Transform of Integral (적분의 라플라스 변환):
( )( ) ( ) ( ) ( ) ( ) ( )⎟⎞⎜⎛==⎟⎟
⎞⎜⎜⎛
⇒= ∫∫ sFdfsFdfsFtf -tt 11 1LLL ττττ( )( ) ( ) ( ) ( ) ( ) ( )⎟
⎠⎜⎝
==⎟⎟⎠
⎜⎜⎝
⇒= ∫∫ sFs
df, sFs
dfsFtf 00
LLL ττττ
1 1Ex. 3 Find the inverse of and( )22
1ω+ss
( )t ωτ 1sin111 11 ⎟
⎞⎜⎛⎞⎛ ∫
( )2221ω+ss
( ) ( )tdss
ts
ωω
τωωτ
ωω
ωωcos11sin1 sin11
20
221
221 −==⎟⎟
⎠
⎞⎜⎜⎝
⎛+
⇒=⎟⎠⎞
⎜⎝⎛
+ ∫−− LL
( )1 sin11 ωttt
⎟⎞
⎜⎛
∫( ) ( ) 320
22221 sincos111
ωω
ωτωτ
ωωttd
ss−=−=⎟⎟
⎠
⎞⎜⎜⎝
⎛+ ∫−L
66.2 Transforms of Derivatives and Integrals..2 Transforms of Derivatives and Integrals.ggODEs ODEs ((도함수와도함수와 적분의적분의 변환변환. . 상미분방정식상미분방정식))
Differential Equations. Initial Value Problems
( ) ( ) ( ) 10 0' ,0 ,''' KyKytrbyayy ===++
• Step 1. Setting up the subsidiary equation (보조방정식의 도출): ( ) ( )rRyY LL == ,
( ) ( )[ ] ( )[ ] ( )sRbYysYaysyYs =+−+−− 00'02
• Step 2 Solution of the subsidiary equation by algebra:
( ) ( ) ( ) ( ) ( )sRyyasYbass +++=++ 0'02
Step 2. Solution of the subsidiary equation by algebra:
Transfer Function (전달함수): ( )2
22
41
21
11
abasbass
sQ−+⎟
⎠⎞
⎜⎝⎛ +
=++
=
solution of the subsidiary equation:
Step 3 Inversion of Y to obtain
42 ⎠⎝
( ) ( ) ( ) ( )[ ] ( ) ( ) ( )sQsRsQyyassY +++= 0'0
( )Yy -1L• Step 3. Inversion of Y to obtain ( )Yy L=
66.2 Transforms of Derivatives and Integrals..2 Transforms of Derivatives and Integrals.ggODEs ODEs ((도함수와도함수와 적분의적분의 변환변환. . 상미분방정식상미분방정식))
Ex 4 Solve the initial value problemEx. 4 Solve the initial value problem
( ) ( ) 10' ,10 ,'' ===− yytyy
Step 1 subsidiary equation
( ) ( ) ( ) 22
22 111 10'0 sYsYysyYs ++=−⇒=−−−
Step 2 transfer function
( ) ( ) ( ) 22 ss
11
2 −=
sQ
( ) ( ) ⎟⎠⎞
⎜⎝⎛ −
−+
−=
−+
−+
=++= 2222221
11
11
11
1111
sssssssQ
sQsY
Step 3 inversion
( ) ( ) tteYty t +=⎟⎞
⎜⎛
⎟⎞
⎜⎛+⎟
⎞⎜⎛== −−−− sinh111 1111 LLLL( ) ( ) tte
sssYty −+=⎟
⎠⎜⎝
−⎟⎠
⎜⎝ −
+⎟⎠
⎜⎝ −
== sinh11 22 LLLL
66.2 Transforms of Derivatives and Integrals..2 Transforms of Derivatives and Integrals.ggODEs ODEs ((도함수와도함수와 적분의적분의 변환변환. . 상미분방정식상미분방정식))
66.2 Transforms of Derivatives and Integrals..2 Transforms of Derivatives and Integrals.ggODEs ODEs ((도함수와도함수와 적분의적분의 변환변환. . 상미분방정식상미분방정식))
66.2 Transforms of Derivatives and Integrals..2 Transforms of Derivatives and Integrals.ggODEs ODEs ((도함수와도함수와 적분의적분의 변환변환. . 상미분방정식상미분방정식))
Ex 2 Derive the formulas in three different waysEx. 2 Derive the formulas in three different ways
( ) cos 22 ωω
+=
sstL
( ) 22sinω
ωω+
=s
tL
66.2 Transforms of Derivatives and Integrals..2 Transforms of Derivatives and Integrals.ggODEs ODEs ((도함수와도함수와 적분의적분의 변환변환. . 상미분방정식상미분방정식))
PROBLEM SET 6.2
HW 24HW: 24
66.3 Unit Step Function. t.3 Unit Step Function. t--Shifting Shifting 66 3 U S ep u c o3 U S ep u c o S gS g((단위계단함수단위계단함수. t. t--이동이동))
Unit Step Function(단위계단함수)의 라플라스 변환
• Unit step function or Heaviside function: ( ) ( )( )⎩
⎨⎧
><
=−atat
atu10
• 단위계단함수의 라플라스 변환 :
( )⎩ > at1
( ){ }s
eatuas−
=−L Prove!
S d Shif i Th (제 2이동정리) Ti Shif i ( 이동)Second Shifting Theorem (제 2이동정리), Time Shifting (t-이동)
( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ){ }sFeatuatfsFeatuatfsFtf as-as −− =−−=−−⇒= 1 , LLL
66.3 Unit Step Function. t.3 Unit Step Function. t--Shifting Shifting 66 3 U S ep u c o3 U S ep u c o S gS g((단위계단함수단위계단함수. t. t--이동이동))
( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ){ }sFeatuatfsFeatuatfsFtf as-as −− =−−=−−⇒= 1 , LLL
( ) ( ) ( )dfedfeesFe assasas
0
)(
0== ∫∫
∞ +−∞ −−− ττττ ττ
( ) ( )
( ) ( ) dtatuatfesFe
dtatfesFe
stas
a
stas
)(=
−=
∫∫∞ −−
∞ −−
( ) ( ) dtatuatfesFe )(0
−−= ∫
66.3 Unit Step Function. t.3 Unit Step Function. t--Shifting Shifting 66 3 U S ep u c o3 U S ep u c o S gS g((단위계단함수단위계단함수. t. t--이동이동))
E 1 W it th f ll i f ti i it t f ti d fi d it t fEx. 1 Write the following function using unit step functions and find its transform.
( )( )( )1
10 2 2⎪⎪
⎨
⎧
<<
<<
= πttt
tf ( ) ( )( )
2 cos21 2
⎪⎪⎩
⎨
>
<<=π
π
tt
tttf
( ) ( )( ) ( ) ( ) ⎟⎞
⎜⎛⎟
⎞⎜⎛
⎟⎞
⎜⎛ 111 2Step 1 단위계단함수의 식:
Step 2 항별 라플라스 변환
( ) ( )( ) ( ) ( ) ⎟⎠⎞
⎜⎝⎛ −+⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ −−−+−−= ππ
21cos
211
21112 2 tuttututtutf
p
( ) ( ) ( ) ( ) 2322
21111
2111
211
21 se
ssstutttut −⎟
⎠⎞
⎜⎝⎛ ++=
⎭⎬⎫
⎩⎨⎧
−⎟⎠⎞
⎜⎝⎛ +−+−=
⎭⎬⎫
⎩⎨⎧ − LL
22
23
222
821
21
821
221
21
21
21 s
esss
tutttutπππππππππ−
⎟⎟⎠
⎞⎜⎜⎝
⎛++=
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎠⎞
⎜⎝⎛ −⎟
⎟⎠
⎞⎜⎜⎝
⎛+⎟⎠⎞
⎜⎝⎛ −+⎟
⎠⎞
⎜⎝⎛ −=
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ − LL
66.3 Unit Step Function. t.3 Unit Step Function. t--Shifting Shifting 66 3 U S ep u c o3 U S ep u c o S gS g((단위계단함수단위계단함수. t. t--이동이동))
( ) 22 11
21
21sin
21cos
se
stuttut
ππππ
−
+−=
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ −⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ −−=
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ − LL
( ) 22
22
2323 11
821
211122
ssss es
esss
esss
ess
fLππππ −−−−
+−⎟⎟
⎠
⎞⎜⎜⎝
⎛++−⎟
⎠⎞
⎜⎝⎛ +++−=⇒
1822 sssssssss +⎠⎝⎠⎝
66.3 Unit Step Function. t.3 Unit Step Function. t--Shifting Shifting 66 3 U S ep u c o3 U S ep u c o S gS g((단위계단함수단위계단함수. t. t--이동이동))
Ex 2 Find the inverse transform f (t) ofEx. 2 Find the inverse transform f (t) of
( )( )2
3
22
2
22 2++
++
+=
−−−
se
se
sesF
sss
ππ
ππ
πt
ssin1
221 =⎟
⎠⎞
⎜⎝⎛
+−L
( )tte
st
s2
21
21
21 1 −−− =⎟⎟
⎠
⎞⎜⎜⎝
⎛
+⇒=⎟
⎠⎞
⎜⎝⎛ LL (제 1이동정리)
( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( )−−+−−+−−=⇒ −− 3322sin111sin1 32 t tuettuttuttf ππ
ππ
( )( ) ( )
( )⎪
⎪⎪⎨
⎧
<<<<−
<<
=3t202t1 sin1t0 0
tπ
π
( )( ) ( ) ( )⎪
⎪⎩ >−
<<−− 3t 3
3t2 032 tet
66.3 Unit Step Function. t.3 Unit Step Function. t--Shifting Shifting 66 3 U S ep u c o3 U S ep u c o S gS g((단위계단함수단위계단함수. t. t--이동이동))
PROBLEM SET 6.3
HW 35HW: 35
66.4 .4 Short Impulses. Dirac’s Delta Function.Short Impulses. Dirac’s Delta Function.ppPartial FractionsPartial Fractions ((짧은짧은 충격충격. . 디랙의디랙의 델타함수델타함수))
Dirac Delta Function의 라플라스 변환Dirac Delta Function의 라플라스 변환
• Dirac delta function or unit impulse function (단위충격함수):
( ) ( )⎧ =∞ at( ) ( )( )⎩
⎨⎧
=−경우 밖의그 0
atatδ
( ) ( ) ( ) ( )kata⎧ +≤≤1( ) ( )
( )( ) ( )atfatkatakatf kkk −=−⇒
⎩⎨⎧ +≤≤=−
→ 0 lim
0
1δ
경우 밖의그
( ) ( ) 111⇒ ∫∫∫
∞+∞
dttdtdttfka
δu(t )와 δ(t ) 관계?
• Dirac delta function의 라플라스 변환:
( ) ( ) 1 100
=−⇒==− ∫∫∫ dtatdtk
dtatfa
k δ
( ) ( ) ( )( )[ ]
ks
k katuatuk
atf
−
+−−−=−
11
1
( )( ) ( )[ ] ( ){ } asks
asskaask eat
kseeee
ksatf −−+−− =−⇒
−=−=−⇒ δLL 11
66.4 .4 Short Impulses. Dirac’s Delta Function.Short Impulses. Dirac’s Delta Function.ppPartial FractionsPartial Fractions ((짧은짧은 충격충격. . 디랙의디랙의 델타함수델타함수))
Ex. 1 Determined the response of the damped mass-spring system under a square wave.
( ) ( ) ( ) 00' ,00 ),2(1)(2'3'' ==−−−==++ yytututryyy ( ) ( ) ( ),),()( yyyyy
( )⎧ ( )
( ) 2121
21
100)1(2)1(
⎪
⎪⎪
⎨
⎧
<<+−
<<
=−−−−
tee
t
ytt
)2(21
21 )2(2)1(2)2()1(⎪
⎪
⎩>
−++− −−−−−−−− teeee tttt
66.4 .4 Short Impulses. Dirac’s Delta Function.Short Impulses. Dirac’s Delta Function.ppPartial FractionsPartial Fractions ((짧은짧은 충격충격. . 디랙의디랙의 델타함수델타함수))
Ex. 2 Find the response of the system with a unit impulse at time t = 1.
( ) ( ) ( ) 00' ,00 ,12'3'' ==−=++ yytyyy δ
상미분방정식: seYsYYs −=++ 232
보조방정식:
( ) ( )( )s
se
ssssesY −−
⎟⎠⎞
⎜⎝⎛
+−
+=
++=
21
11
21
( ) ( ) ( )( ) ( ) ( )⎩
⎨⎧
><<
== −−−−−
1t1t0 0
1211
tt eeYty L
( )( ) ⎠⎝
( ) ( ) ( )⎩ >− 1t ee
( ) ( ) ( ){ }sFeatuatf as- −=−− 1 L( ) ( ) ( ){ }f
66.4 .4 Short Impulses. Dirac’s Delta Function.Short Impulses. Dirac’s Delta Function.ppPartial FractionsPartial Fractions ((짧은짧은 충격충격. . 디랙의디랙의 델타함수델타함수))
PROBLEM SET 6.4
HW 16 (b)HW: 16 (b)
66.5 .5 Convolution. Integral Equations Convolution. Integral Equations 66 55 Co o u o eg a qua o sCo o u o eg a qua o s((합성곱합성곱. . 적분방정식적분방정식))
( ) ( ) ( ) ( ) ( )( )1 , ex)
? ==
=⇒≠
gefgfgffg
t
LLLLLL -1
66.5 .5 Convolution. Integral Equations Convolution. Integral Equations 66 55 Co o u o eg a qua o sCo o u o eg a qua o s((합성곱합성곱. . 적분방정식적분방정식))
( ) ( ) ( ) ( ) ( )( )1 , ex)
? ==
=⇒≠
gefgfgffg
t
LLLLLL -1
Convolution (합성곱):
Properties of Convolution
( )( ) ( ) ( ) τττ dtgftgft
−=∗ ∫0
• Commutative law:
• Distributive law:
fggf ∗=∗
( ) 2121 gfgfggf ∗+∗=+∗
• Associative law:
•
( ) ( )vgfvgf ∗∗=∗∗
000 =∗=∗ ff
• 합성곱의 특이성질:
Convolution Theorem: ( ) ( ) ( )gfgf LLL =∗
ff ≠∗1
66.5 .5 Convolution. Integral Equations Convolution. Integral Equations 66 55 Co o u o eg a qua o sCo o u o eg a qua o s((합성곱합성곱. . 적분방정식적분방정식))
( )( ) ( ) ( ) τττ dtgftgft
−=∗ ∫Ex. 1 Let . Find( ) ( )[ ]sassH −= 1 ( )th
⎞⎛⎞⎛
( )( ) ( ) ( )gfgf ∫0
( ) ( ) ( )gfgf LLL =∗
( ) ( )1111
11 ,1 11
==∗=⇒
=⎟⎠⎞
⎜⎝⎛=⎟
⎠⎞
⎜⎝⎛
−
∫
−−−
att
aat
at
edeeth
se
as
ττ
LL
( ) ( )111 0
−=⋅=∗=⇒ ∫ ea
deeth τ
66.5 .5 Convolution. Integral Equations Convolution. Integral Equations 66 55 Co o u o eg a qua o sCo o u o eg a qua o s((합성곱합성곱. . 적분방정식적분방정식))
( )( ) ( ) ( ) τττ dtgftgft
−=∗ ∫Ex. 2 Let . Find( ) ( )222
1ws
sH+
= ( )th( )( ) ( ) ( )gfgf ∫
0
( ) ( ) ( )gfgf LLL =∗
66.5 .5 Convolution. Integral Equations Convolution. Integral Equations 66 55 Co o u o eg a qua o sCo o u o eg a qua o s((합성곱합성곱. . 적분방정식적분방정식))
( )( ) ( ) ( ) τττ dtgftgft
−=∗ ∫Ex. 2 Let . Find( ) ( )222
1ws
sH+
= ( )th
⎞⎛
( )( ) ( ) ( )gfgf ∫0
( ) ( ) ( )gfgf LLL =∗
( ) ∗⇒
=⎟⎠⎞
⎜⎝⎛
+−
∫ dtwtwtth
wwtws
t
)(ii1sinsin
/)(sin122
1L
( )
[ ]+−=
−=∗=⇒
∫
∫
dwwtw
dtwwwww
th
t
coscos2
1
)(sinsin
2
02
ττ
τττ
⎥⎦⎤
⎢⎣⎡ +−=
=
∫
wwwt
w
wt
r
sincos2
1
2
02
0
ττ
⎥⎦⎤
⎢⎣⎡ +−=
wwtwtt
wsincos
21 2
66.5 .5 Convolution. Integral Equations Convolution. Integral Equations 66 55 Co o u o eg a qua o sCo o u o eg a qua o s((합성곱합성곱. . 적분방정식적분방정식))
Ex. 4 In an undamped mass-spring system, resonance occurs if the frequency of the driving force equals the natural frequency of the system.
( ) ( ) 00'00sin'' 2+ tK ( ) ( ) 00' ,00 ,sin'' 020 ===+ yytKyy ϖϖ
( ) ( ) ( ) ( ) ( ) ( ) ( )0'0'' 0' 2 fsffsf, ffsf −−=−= LLLL
66.5 .5 Convolution. Integral Equations Convolution. Integral Equations 66 55 Co o u o eg a qua o sCo o u o eg a qua o s((합성곱합성곱. . 적분방정식적분방정식))
Ex. 4 In an undamped mass-spring system, resonance occurs if the frequency of the driving force equals the natural frequency of the system.
( ) ( ) 00'00sin'' 2+ tK ( ) ( ) 00' ,00 ,sin'' 020 ===+ yytKyy ϖϖ
( ) ( ) ( ) ( ) ( ) ( ) ( )0'0'' 0' 2 fsffsf, ffsf −−=−= LLLL
KY 2220ϖ
=
( ) ( )tttKty
sY
0002
220
2
sincos2
)(
ϖϖϖ
ϖ
+−=⇒
+
( ) ( )y 000202ϖ
66.5 .5 Convolution. Integral Equations Convolution. Integral Equations 66 55 Co o u o eg a qua o sCo o u o eg a qua o s((합성곱합성곱. . 적분방정식적분방정식))
Integral Eq ation (적분방정식)Integral Equation (적분방정식):
미지의 함수 y(t )가 적분기호 안 또는 밖에 나타나는 방정식
Ex 6 Solve the Volterra integral equation of the second kindEx. 6 Solve the Volterra integral equation of the second kind.
( ) ( ) ( ) tdtytyt
=−− ∫ τττ sin0
합성곱을 이용
0
ttyy =∗− sin
라플라스 변환 ( ) ( ) 221
11
sssYsY =
+−
32( ) ( )
6 111 3
424
2 tttysss
ssY +=⇒+=+
=
66.5 .5 Convolution. Integral Equations Convolution. Integral Equations 66 55 Co o u o eg a qua o sCo o u o eg a qua o s((합성곱합성곱. . 적분방정식적분방정식))
PROBLEM SET 6.5
HW 25HW: 25
66.6 Differentiation and Integration of.6 Differentiation and Integration ofT ansfo ms ODEs ith Va iable CoefficientsT ansfo ms ODEs ith Va iable CoefficientsTransforms. ODEs with Variable CoefficientsTransforms. ODEs with Variable Coefficients((변환의변환의 미분과미분과 적분적분. . 변수계수의변수계수의 상미분방정식상미분방정식))
( ) ( ) ( )∫∞
−==0
dttfefsF stL
( ) ( ) ( )∫∞
− −=−==⇒0
0
' tfdtttfedsdFsF st L
0
( )( ) ( ) ( ){ } ( )ttfsFsFttf −=−=⇒ − ' ,' 1LL
66.6 Differentiation and Integration of.6 Differentiation and Integration ofT ansfo ms ODEs ith Va iable CoefficientsT ansfo ms ODEs ith Va iable CoefficientsTransforms. ODEs with Variable CoefficientsTransforms. ODEs with Variable Coefficients((변환의변환의 미분과미분과 적분적분. . 변수계수의변수계수의 상미분방정식상미분방정식))
Ex 1 Derive the following three formulasEx. 1 Derive the following three formulas.
( )( ) ( )Fttf 'L ( )( ) ( )sFttf '−=L
( ) 22sinβ
ββ+
=s
tL ( ) ( )222
2sinβ
ββ+
=s
sttL미분에 의하여
β ( )β+s
( )222sin
2 ββ
β=⎟⎟
⎠
⎞⎜⎜⎝
⎛ sttL ( )222 ββ +⎠⎝ s
66.6 Differentiation and Integration of.6 Differentiation and Integration ofT ansfo ms ODEs ith Va iable CoefficientsT ansfo ms ODEs ith Va iable CoefficientsTransforms. ODEs with Variable CoefficientsTransforms. ODEs with Variable Coefficients((변환의변환의 미분과미분과 적분적분. . 변수계수의변수계수의 상미분방정식상미분방정식))
( ) 22cosβ
β+
=s
stL ( ) ( )( ) ( )222
22
222
222 2cosβ
β
β
ββ+
−=
+
−+−=
s
s
s
ssttL미분에 의하여
β ( ) ( )ββ ++ ss
( )( ) ( )
( )222
2222
22222
22 1sin1cos βββ
βββ
β +±−=±
−=⎟⎟
⎠
⎞⎜⎜⎝
⎛±
ssstttL ( ) ( )22222222 ββββ
ββ
+++⎟⎠
⎜⎝ sss
( ) ( ) ( ) ( )tfsdsFsdsFtf=⎬
⎫⎨⎧
⇒=⎬⎫
⎨⎧
∫∫∞
−∞
~~~~ 1LL
Integration of Transforms (변환의 적분):
( ) ( ) ( ) ( )t
sdsFsdsFt ss
=⎭⎬
⎩⎨⇒=
⎭⎬
⎩⎨ ∫∫ LL
66.6 Differentiation and Integration of.6 Differentiation and Integration ofT ansfo ms ODEs ith Va iable CoefficientsT ansfo ms ODEs ith Va iable CoefficientsTransforms. ODEs with Variable CoefficientsTransforms. ODEs with Variable Coefficients((변환의변환의 미분과미분과 적분적분. . 변수계수의변수계수의 상미분방정식상미분방정식))
222 ⎞⎛Ex. 2 Find the inverse transform of .2
22
2
2ln1ln
ss
sωω +
=⎟⎟⎠
⎞⎜⎜⎝
⎛+
( )( ) 22d222 +⎞⎛ 미분 ( )( ) 222222 22lnln
ss
ssss
dsd
−+
=−+ω
ω2
22
2
2ln1ln
ss
sωω +
=⎟⎟⎠
⎞⎜⎜⎝
⎛+
미분
Case 1) 변환의 미분이용
( ) ( ) ( )( ) ( )ttftss
sssF
sfsF −=−=⎟
⎠⎞
⎜⎝⎛ −
+=⇒⎟⎟
⎠
⎞⎜⎜⎝
⎛+== −− 2cos222' 1ln 222
112
2ω
ωω
LLLsss ⎠⎝ +⎠⎝ ω
( ) ( )tt
tf ωcos12 −=∴
22Case 2) 적분이용 ( ) ( ) ( ) ( )1cos2 22 1
22 −==⇒−+
= − tGtgss
ssG ωω
L
( ) ( ) ( )tgdGω 12~~1l 12
1 ⎟⎞
⎜⎛
⎟⎞
⎜⎛
⎟⎞
⎜⎛
∫∞
LL ( ) ( ) ( )ttt
tgsdsGs s
ωω cos12~~1ln 12
1 −=−=⎟⎟⎠
⎜⎜⎝
=⎟⎟⎠
⎜⎜⎝
⎟⎟⎠
⎜⎜⎝+ ∫−− LL
66.6 Differentiation and Integration of.6 Differentiation and Integration ofT ansfo ms ODEs ith Va iable CoefficientsT ansfo ms ODEs ith Va iable CoefficientsTransforms. ODEs with Variable CoefficientsTransforms. ODEs with Variable Coefficients((변환의변환의 미분과미분과 적분적분. . 변수계수의변수계수의 상미분방정식상미분방정식))
Variable Coefficient (변수계수)를 가진 상미분방정식Variable Coefficient (변수계수)를 가진 상미분방정식
( ) ( )[ ]0'dsdYsYysY
dsdty −−=−−=L
( ) ( ) ( )[ ] ( )020'0'' 22 ydsdYssYysyYs
dsdty +−−=−−−=L
Ex 3 Laguerre’s Equation Laguerre PolynomialsEx. 3 Laguerre s Equation. Laguerre Polynomials.
( ) ( ), , , nnyytty 210 0'1'' ==+−+
( ) ( ) ( ) ( )22 010002 ++⇒+⎟⎞
⎜⎛+⎥
⎤⎢⎡ + YsndYs snYdYsYysYydYssY ( ) ( ) ( ) ( )
( )111
01 0002
−=⇒⎟
⎞⎜⎛ +
=−+
=⇒
=−++⇒=+⎟⎠
⎜⎝
−−−−+⎥⎦⎢⎣+−−
nsYdsnndssndY
Ysnds
s-snYds
sYysYyds
ssY
( )12
1 +=⇒⎟
⎠⎜⎝
−−
=−=⇒ nsYds
ssds
s-sY
( ) ( ) ( )⎪⎨⎧ =
== −0 1
1 de n,
Ytl ntL( ) ( ) ( )⎪⎩⎨ =
== − ,2 ,1 ,!
netdtd
neYtl tn
nn L
66.6 Differentiation and Integration of.6 Differentiation and Integration ofT ansfo ms ODEs ith Va iable CoefficientsT ansfo ms ODEs ith Va iable CoefficientsTransforms. ODEs with Variable CoefficientsTransforms. ODEs with Variable Coefficients((변환의변환의 미분과미분과 적분적분. . 변수계수의변수계수의 상미분방정식상미분방정식))
PROBLEM SET 6.6
HW 15HW: 15
66.7 Systems of ODEs.7 Systems of ODEs ((연립연립상미분방정식상미분방정식))66 Sys e s o O sSys e s o O s ((연립연립상미분방정식상미분방정식))Ex. 3 The mechanical system consists of two bodies of mass 1 of three
springs of the same spring constant k and of negligibly small massesof the springs Also damping is assumed to be practically zeroof the springs. Also damping is assumed to be practically zero.
( ) ( ) ( ) ( ) kyyyy 300 ,100 '2
'121 =−===
66.7 Systems of ODEs.7 Systems of ODEs ((연립연립상미분방정식상미분방정식))66 Sys e s o O sSys e s o O s ((연립연립상미분방정식상미분방정식))Ex. 3 The mechanical system consists of two bodies of mass 1 of three
springs of the same spring constant k and of negligibly small massesof the springs Also damping is assumed to be practically zeroof the springs. Also damping is assumed to be practically zero.
( ) ( ) ( ) ( ) kyyyy 300 ,100 '2
'121 =−===
( )12112 3 YYkkYksYs −+−=−−
( )1211 '' yykkyy −+−= 라플라스 변환
지배방정식:
( ) 21222 3 kYYYkksYs −−−=+−( ) 2122 '' kyyyky −−−=
Cramer의 법칙
( )( ) ( ) kskskksksY 3323 2+=
−+++=
또는 소거법 적용
( )
( )( ) ( ) kskskksksY
kskskksY
3323
32
2
222221
=−++−
=
++
+=
−+=
( ) ( )
( ) ( ) tktkYty
tktkYty
3sincos
3sincos
21
2
11
1
−==
+==
−
−
L
L 역변환
( )( ) ( )( ) kskskks
Y32
222222 +−
+=
−+=( ) ( )22
66.7 Systems of ODEs.7 Systems of ODEs ((연립연립상미분방정식상미분방정식))66 Sys e s o O sSys e s o O s ((연립연립상미분방정식상미분방정식))
PROBLEM SET 6.7
HW 22HW: 22