Bong-Kee Lee School of Mechanical Systems Engineering
Chonnam National University
Engineering Mathematics I
2. Second-Order Linear ODEs
School of Mechanical Systems Engineering Engineering Mathematics I
2.1 Homogeneous Linear ODEs of 2nd Order
2계 선형상미분방정식
– 표준형(standard form)
xryxqyxpy ''' nonhomogeneous
0''' yxqyxpy homogeneous
0'''
0'1
''
cos25''
example
2
yyy
yyx
y
xeyy x2nd order nonhomogeneous linear ODE
2nd order homogeneous linear ODE
2nd order homogeneous nonlinear ODE
School of Mechanical Systems Engineering Engineering Mathematics I
2.1 Homogeneous Linear ODEs of 2nd Order
중첩의 원리(superposition principle) 또는 선형성의 원리(linearity principle)
– 주어진 해에 어떤 상수를 더하거나 곱함으로써 추가적인 해를 얻을 수 있음
– (정리) 제차 선형상미분방정식에 대하여, 어떤 열린 구간 I에서 두 개 해의 일차결합은 다시 구간 I에서 주어진 미분방정식의 해가 된다. 특히 그러한 방정식에 대해서는 해들의 합과 상수곱도 다시 해가 된다.
2211
21 &
0'''
ycycy
yyyy
qypyy
School of Mechanical Systems Engineering Engineering Mathematics I
2.1 Homogeneous Linear ODEs of 2nd Order
중첩의 원리(superposition principle) 또는 선형성의 원리(linearly principle)
– 비제차 선형상미분방정식 및 비선형 상미분방정식에는 적용되지 않음
0sin2cos7.4sin2cos7.4
sin2cos7.4''sin2cos7.4
sin2cos7.4
0sinsinsin''sin2
0coscoscos''cos1
sin&cos0''example
xxxx
xxxx
xxy
xxxx
xxxx
xyxyyy
2cos22cos2cos12''cos12
cos12
1sin1sinsin1''sin12
1cos1coscos1''cos11
sin1&cos11''example
xxxx
xy
xxxx
xxxx
xyxyyy
22222
22222
2
4'''
0'11''12
022'''1
1&0'''example
xxxxxxy
x
xxxxxx
yxyxyyy
School of Mechanical Systems Engineering Engineering Mathematics I
2.1 Homogeneous Linear ODEs of 2nd Order
초기값 문제(initial value problem, IVP)
– 2계 제차 상미분방정식의 일반해(general solution)
– 2계 제차 상미분방정식에는 두 개의 초기조건이 요구됨
– 2계 제차 상미분방정식의 초기값 문제
2211 yCyCy
1000 ' & KxyKxy
1000 '&with
0'''
KxyKxy
yxqyxpy
School of Mechanical Systems Engineering Engineering Mathematics I
2.1 Homogeneous Linear ODEs of 2nd Order
초기값 문제(initial value problem, IVP)
xxy
CyxCxCy
CyxCxCy
xCxCyxyxy
yyyy
sin5.0cos0.3
5.00'cossin'
0.30sincosstep 2nd
sincossin&cosstep1st
5.00',0.30,0''example
221
121
21
general solution
particular solution
School of Mechanical Systems Engineering Engineering Mathematics I
2.1 Homogeneous Linear ODEs of 2nd Order
기저(basis) 또는 기본시스템(fundamental system)
– 일반해를 구성하는 해(y1, y2)는 구간 I에서 1차 독립(linearly independent)이어야 함
– 비례(proportional)
• 주어진 두 초기조건을 만족시키는 해를 정의할 수 없음
– 1차 독립
– 1차 종속(linearly dependent)
0const 21 kkyy
00 212211 kkykyk
School of Mechanical Systems Engineering Engineering Mathematics I
2.1 Homogeneous Linear ODEs of 2nd Order
기저(basis) 또는 기본시스템(fundamental system)
– 기저: 1차 독립인 해의 쌍(pair)
xx
xx
x
x
x
xxxx
xxxx
xx
eey
CCy
CCy
eCeCyCyCy
ee
e
y
y
eeeeyy
eeeeyy
eyey
yyyy
42
20'2
601
const
0''''2
0''''1
?&
20',60,0''example
21
21
212211
2
2
1
22
11
21
general solution
particular solution
School of Mechanical Systems Engineering Engineering Mathematics I
2.1 Homogeneous Linear ODEs of 2nd Order
차수 축소(reduction of order)
– 하나의 해를 알고 있을 때, 두 번째 독립해를 1계 상미분방정식을 이용하여 구할 수 있음
Udxyuyyey
U
pdxyUdxpy
y
U
dUUp
y
yU
UuUupyyuyu
qypyyupyyuyu
uyquyyupuyyuyu
yxqyxpy
uyyuyuuyyuyuyuyy
uyyuyyuyyy
yyxqyxpy
pdx
1122
1
1
1
1
1
1
111
111111
111111
11111112
11212
1
&1
ln2ln'2
0'2
'
''','0'2'''
0''''2'''
0''''''2''
0'''
''''2''''''''''''''
''''
solution,known e with th0'''
School of Mechanical Systems Engineering Engineering Mathematics I
2.1 Homogeneous Linear ODEs of 2nd Order
차수 축소(reduction of order)
1ln1
ln
1111lnln21lnln
2
1
1202':'
0'2''0''2''
'2''''''''''
? then , intuition,by
0'''example
2
222
2
2
22
12
21
2
xxuxyx
xu
xxx
x
dx
duv
x
xxxv
dxxx
dxxx
x
v
dvvxvxxuv
uxuxxuxuxuxuxuxx
uxuuuxuyuxuy
uxuyyy
yxy
yxyyxx
separating variables
School of Mechanical Systems Engineering Engineering Mathematics I
2.2 Homogeneous Linear ODEs: Const. Coffs.
상수계수를 갖는 2계 제차 선형상미분방정식
– 특성방정식(characteristic equation), 보조방정식(auxiliary equation)
0'''
'''
byayy
xryxqyxpy
0''':
''&' assumingby
22
2
xxxx
xxx
ebabeeaebyayy
eyeyey
02 baxx
eyey 21
2121 &&
School of Mechanical Systems Engineering Engineering Mathematics I
2.2 Homogeneous Linear ODEs: Const. Coffs.
특성방정식의 근에 따른 세 경우
– (a)>0: 두 개의 서로 다른 실근(two real roots)
– (a)=0: 실 이중근(real double root)
– (a)<0: 공액 복소근(complex conjugate roots)
2
4
0
2
2,1
2
bab
ba
(a)
School of Mechanical Systems Engineering Engineering Mathematics I
2.2 Homogeneous Linear ODEs: Const. Coffs.
특성방정식의 근에 따른 세 경우(I)
– 두 개의 서로 다른 실근(two real roots)
xxeCeCy 21
2121 & general solution
xx
xx
eey
CCCCy
CCy
eCeCy
yyyyy
2
21
21
21
2
21
2
3
3,1520'
40
solution particular
solution general
2,102equation sticcharacteri
50',40,02'''example
School of Mechanical Systems Engineering Engineering Mathematics I
2.2 Homogeneous Linear ODEs: Const. Coffs.
특성방정식의 근에 따른 세 경우(II)
– 실 이중근(real double root)
xx xeCeCya 21212
general solution
x
xx
x
exy
CC
CCy
exCCeCxy
Cy
exCCy
yyyyy
5.0
21
12
5.0
21
5.0
2
1
5.0
21
2
23
2,3
5.35.00'
5.0'
30
solution particular
solution general
5.0025.0equation sticcharacteri
5.30',30,025.0'''example
School of Mechanical Systems Engineering Engineering Mathematics I
2.2 Homogeneous Linear ODEs: Const. Coffs.
특성방정식의 근에 따른 세 경우(III)
– 공액 복소근(complex conjugate roots)
xeCxeCyia
ia axax sincos
2,
2
2/
2
2/
121
general solution
xey
CC
Cy
xeCxeCxy
Cy
xeCxeCy
i
yyyyy
x
xx
xx
3sin
1,0
330'
3cos33sin2'
00
solution particular
3sin3cossolution general
32.0
004.94.0equation sticcharacteri
30',00,004.9'4.0''example
2.0
21
2
2.0
2
2.0
2
1
2.0
2
2.0
1
2
School of Mechanical Systems Engineering Engineering Mathematics I
2.2 Homogeneous Linear ODEs: Const. Coffs.
특성방정식의 근에 따른 세 경우(III)
– 공액 복소근(complex conjugate roots)
xeCxeCyia
ia axax sincos
2,
2
2/
2
2/
121
xixeeeeey
xixeeeeey
iaiaa
abbaba
baaba
xa
xix
axia
x
xa
xix
axia
x
sincos
sincos
22
2
2
4
44404
2
40
222
2
222
1
22
2,1
2222
2
2,1
2
2
1
2122112
1,
2
1yy
iYyyY
sincos iei
Euler formula
School of Mechanical Systems Engineering Engineering Mathematics I
2.3 Differential Operators
연산자(operator)
– 함수를 다른 함수로 바꾸는 변환을 의미
– 미분연산자(differential operator)
– 항등연산자(identity operator)
– 선형연산자(linear operator)
'yydx
dy
dx
d DD
yy I
byayyybyayybay
ba
'''22
2
IDDIDD
IDD
L
L
School of Mechanical Systems Engineering Engineering Mathematics I
2.4 Modeling: Free Oscillations
자유진동(free oscillation): 질량-용수철 시스템(mass-spring system)
''2
2
mydt
ydmmaF Newton’s second law
kyF Hooke’s law
000 ksmgksWF
static equilibrium
0''
''
kymy
mykyF
undamped system
School of Mechanical Systems Engineering Engineering Mathematics I
2.4 Modeling: Free Oscillations
자유진동(free oscillation): 질량-용수철 시스템(mass-spring system)
– 비감쇠 시스템(undamped system) • 조화진동(harmonic oscillation)
A
BBACtCty
m
ktBtAtykymy
122
0
000
tan, ,cos
,sincos0''
amplitude (진폭)
phase angle (위상각)
2
0
natural frequency (고유주파수, Hz)
(a)
(b)
(c)
1)c(
0)b(
1)a(
1,10
B
B
B
A
School of Mechanical Systems Engineering Engineering Mathematics I
2.4 Modeling: Free Oscillations
자유진동(free oscillation): 질량-용수철 시스템(mass-spring system)
– 감쇠 시스템(damped system)
'cyF damping force
0'''
'''
kycymy
mycykyF
damped system
homogeneous 2nd order linear ODE (case I) overdamping (case II) critical damping (case III) underdamping
m
mkcckcm
2
40
22
School of Mechanical Systems Engineering Engineering Mathematics I
2.4 Modeling: Free Oscillations
자유진동(free oscillation): 질량-용수철 시스템(mass-spring system)
– 감쇠 시스템(damped system): 과감쇠(overdamping)
m
mkc
m
ceCeCty
mkckycymy
tt
2
4&
2 ,
040'''
2
21
2
School of Mechanical Systems Engineering Engineering Mathematics I
2.4 Modeling: Free Oscillations
자유진동(free oscillation): 질량-용수철 시스템(mass-spring system)
– 감쇠 시스템(damped system): 임계감쇠(critical damping)
m
cetCCty
mkckycymy
t
2 ,
040'''
21
2
School of Mechanical Systems Engineering Engineering Mathematics I
2.4 Modeling: Free Oscillations
자유진동(free oscillation): 질량-용수철 시스템(mass-spring system)
– 감쇠 시스템(damped system): 저감쇠(underdamping)
m
cmk
m
ctCetBtAety
mkckycymy
tt
2
4*&
2 ,*cos*sin*cos
040'''
2
2
frequency
2
1
2
4
2
* 2
m
cmk
as c→0
2
1
22
* 0
m
k
School of Mechanical Systems Engineering Engineering Mathematics I
2.4 Modeling: Free Oscillations
자유진동(free oscillation): 질량-용수철 시스템(mass-spring system)
– 감쇠 시스템(damped system)
c↓
School of Mechanical Systems Engineering Engineering Mathematics I
2.4 Modeling: Free Oscillations
자유진동(free oscillation): 질량-용수철 시스템(mass-spring system)
– 감쇠 시스템(damped system)
c=10
c=5
c=1
c=0
School of Mechanical Systems Engineering Engineering Mathematics I
2.5 Euler-Cauchy Equations
오일러-코시 방정식(Euler-Cauchy equation)
– 2계 제차 상미분방정식의 한 형태
0'''2 byaxyyx
01
01
011''
'
, assumingby
2
122
2
1
bmam
bammm
bxmxaxxmmxxmmy
mxy
xxy
mmm
m
m
m
characteristic equation
School of Mechanical Systems Engineering Engineering Mathematics I
2.5 Euler-Cauchy Equations
오일러-코시 방정식(Euler-Cauchy equation)
– (case I) 서로 다른 두 실근
– (case II) 이중근
– (case III) 공액복소근
21
21
21
2 ,01
mmxCxCy
mmbmam
mmm xxCCxxCxCy
mbmam
lnln
01
2121
2
xBxAxy
imbmam
lnsinlncos
012
School of Mechanical Systems Engineering Engineering Mathematics I
2.5 Euler-Cauchy Equations
오일러-코시 방정식(Euler-Cauchy equation)
xxyxuux
UUx
UuU
ux
uxuuxxummxuux
yxumymxuyux
uymxymyxyxumyuyux
ymuyyuxmuyyuyuxuyyuyuyy
uyyuyy
uyyymxmxyxy
ma
bmaamba
baambmamxybyaxyyx
m
mm
m
lnln'1
01
':'
0'1
''0'''0'21'2''
0'21'2''
0'21'''21''2''
0''21''''2''''''2''''''
''''
',
4
1&211
2
1 then ,041 if
2
411010'''example
2
22
11
1
1
2
1
2
11
2
111
2
2
2
11111
2
1112
112
121
11
11
2
22
2
22
School of Mechanical Systems Engineering Engineering Mathematics I
2.5 Euler-Cauchy Equations
오일러-코시 방정식(Euler-Cauchy equation)
xBxAxy
YCYCy
xxyyi
Y
xxyyY
xixxexexxxxy
xixxexexxxxy
xyxyimba
baambmamxybyaxyyx
xiixii
xiixii
ii
m
lnsinlncos
lnsin2
1
lncos2
1
lnsinlncos
lnsinlncos
& then ,041 if
2
411010'''example
2211
212
211
lnln
2
lnln
1
21
2
2
22
School of Mechanical Systems Engineering Engineering Mathematics I
2.6 Existence and Uniqueness of Solutions
해의 존재성과 유일성
– 연속적인 가변변수를 가지는 제차 선형미분방정식의 해
1000 ',with
0'''
KxyKxy
yxqyxpy
2211 ycycy
solution ?
Theorem 1
Theorem 4
School of Mechanical Systems Engineering Engineering Mathematics I
2.6 Existence and Uniqueness of Solutions
해의 존재성과 유일성
. interval on the
solution unique a has problem valueinitial ingcorrespond then the
,in is and intervalopen someon functions continuous are and If
1. Theorem
0
I
xy
IxIxqxp
constants.
suitable are , and on ODE theof solutions of basisany is , where
form theof is on solution every then ,
intervalopen someon and tscoefficien continuous has ODEgiven theIf
4. Theorem
2121
2211
CCIyy
xyCxyCxY
IxYyI
xqxp
School of Mechanical Systems Engineering Engineering Mathematics I
2.6 Existence and Uniqueness of Solutions
해의 존재성과 유일성
– Wronski 행렬식(determinant) 또는 Wronskian
''''
, 1221
21
21
21 yyyyyy
yyyyW
dependentlinearly :,0, 2121 yyyyW
xx
yy
yyW
xyxy
yy
22
21
21
21
2
sincos
''
sin,cos
0''example
xxx
xx
exeex
yy
yyW
xeyey
yyy
222
21
21
21
1
''
,
0'2''example
School of Mechanical Systems Engineering Engineering Mathematics I
2.7 Nonhomogeneous ODEs
2계 비제차 상미분방정식
– 일반해 • 제차 상미분방정식의 일반해 + 비제차방정식의 어떤 해
– 제차 방정식의 해와 비제차 방정식의 해 사이의 관계 • 비제차 방정식의 두 해의 차는 제차 방정식의 해
• 비제차 방정식의 해와 제차 방정식의 해의 합은 비제차 방정식의 해
0''' xryxqyxpy
xryxqyxpyxy
yxqyxpyyCyCxy
xyxyxy
pppp
hhhh
ph
'''
0'''1211
School of Mechanical Systems Engineering Engineering Mathematics I
2.7 Nonhomogeneous ODEs
미정계수법(method of undetermined coefficients)
– 비제차 방정식의 어떤 해를 구하기 위한 방법
– 일반적인 방법(매개변수 변환법, method of variation of parameter; Chap. 2.10)보다 간단하게 적용이 가능
– 상수계수를 가지는 선형상미분방정식에 적합
xke xCe
xr xyp
,1,0 nkxn01
1
1 KxKxKxK n
n
n
n
xkxk sinor cos xMxK sincos
xkexke xx sinor cos xMxKe x sincos
School of Mechanical Systems Engineering Engineering Mathematics I
2.7 Nonhomogeneous ODEs
미정계수법(method of undetermined coefficients)
– 선택 규칙(I): 기본 규칙(basic rule) • 비제차 상미분방정식의 r(x)가 표의 함수 중 하나라면, 같은 줄의
yp를 선택하고, yp와 그 도함수를 비제차 상미분방정식에 도입하여 미정계수를 결정함
002.0001.0002.0,0,001.0
22
001.0''
2''
2'
~001.0solution particular:step 2nd
sincos0''solution shomogeneou:step1st
5.10',00 with 001.0''example
2
012
021
2
201
2
22
2
2
12
01
2
2
2
2
xyKKK
KKxKxKKxKxKK
xyy
Ky
KxKy
KxKxKy
kxxxr
xBxAyyy
yyxyy
p
pp
p
p
p
n
h
School of Mechanical Systems Engineering Engineering Mathematics I
2.7 Nonhomogeneous ODEs
미정계수법(method of undetermined coefficients)
002.0001.0sin5.1cos002.0
5.10'
002.0cossin'
002.00002.00
conditions initial:step 3rd
002.0001.0sincos
002.0001.0&sincos
2
2
2
xxxy
By
xxBxAy
AAy
xxBxAyyy
xyxBxAy
ph
ph
xy
xyh
xyp
School of Mechanical Systems Engineering Engineering Mathematics I
2.7 Nonhomogeneous ODEs
미정계수법(method of undetermined coefficients)
– 선택 규칙(II): 변형 규칙(modification rule) • 만약 yp로 선택된 항이 비제차 상미분방정식에 대응하는 제차 상
미분방정식의 해가 된다면, 선택된 yp에 x(또는 x2)를 곱함
x
p
x
ppp
x
p
x
p
xx
p
xx
x
h
x
exy
CCxxxCxxC
eyyy
exxCy
exxCy
CeeCxy
keexr
exCCyyyy
yyeyyy
5.12
222
5.1
5.12
5.12
5.15.12
5.1
5.1
21
5.1
5
51025.25.12325.262
1025.2'3''
25.262''
5.12'
~10solution particular:step 2nd
025.2'3''solution shomogeneou:step1st
00',10 with 1025.2'3''example
School of Mechanical Systems Engineering Engineering Mathematics I
2.7 Nonhomogeneous ODEs
미정계수법(method of undetermined coefficients)
x
xx
x
ph
x
p
x
h
exxy
CCy
exxCCexCy
Cy
exxCCyyy
exyexCCy
5.12
22
5.12
21
5.1
2
1
5.12
21
5.125.1
21
55.11
5.105.10'
55.110'
10
conditions initial:step 3rd
5
5&
xy
xyh
xyp
School of Mechanical Systems Engineering Engineering Mathematics I
2.7 Nonhomogeneous ODEs
미정계수법(method of undetermined coefficients)
– 선택 규칙(III): 합 규칙(sum rule) • 만약 r(x)가 표의 함수들의 합이라면, 이에 대응하는 함수들의 합
으로 yp를 선택함
xey
KMC
xxeyyyxMxKCey
xMxKyCey
xxyeyyyy
xMxKkexxexr
xBxAeyyyy
yyxxeyyy
x
p
x
ppp
x
p
p
x
p
p
x
pppp
xx
x
h
x
10sin216.0
0,2,16.0
10sin19010cos405'2''10sin10cos
10sin10cos&
10sin19010cos40~&~
sincos~10sin19010cos40solution particular:step 2nd
2sin2cos05'2''solution shomogeneou:step1st
08.400',16.00 with 10sin19010cos405'2''example
5.0
5.05.0
2
5.0
1
1
5.0
121
5.0
5.0
School of Mechanical Systems Engineering Engineering Mathematics I
2.7 Nonhomogeneous ODEs
미정계수법(method of undetermined coefficients)
xy
xyh
xyp
xexey
BBy
xexBexBey
AAy
xexBxAeyyy
xeyxBxAey
xx
xxx
xx
ph
x
p
x
h
10sin216.02sin10
1008.402008.020'
cos2008.02cos22sin'
016.016.00
conditions initial:step 3rd
10sin216.02sin2cos
10sin216.0&2sin2cos
5.0
5.0
5.0
5.0
School of Mechanical Systems Engineering Engineering Mathematics I
2.8 Modeling: Forced Oscillations. Resonance
강제운동(forced motion)
– 외력(external force)이 존재하는 경우에 대한 운동
→ 비제차 상미분방정식
– 자유운동(free motion): 외력이 존재하지 않는 경우
→ 제차 상미분방정식
0''' kycymy
trkycymy '''
system xr xy
homogeneous ODE
external force response
free motion
forced motion
School of Mechanical Systems Engineering Engineering Mathematics I
2.8 Modeling: Forced Oscillations. Resonance
주기적인 외력이 가해지는 경우
tFtrkycymy cos''' 0
ph yyy
tCyh 0cos
tbtayp sincos
22220
2222
2
0
cmk
cFb
cmk
mkFa
22222
0
20
22222
0
2
22
00
cm
cFb
cm
mFa
2
00 mkm
k
School of Mechanical Systems Engineering Engineering Mathematics I
2.8 Modeling: Forced Oscillations. Resonance
주기적인 외력이 가해지는 경우
– 비감쇠 강제진동(c=0)
tm
FtCy
tbtatCyyy ph
coscos
sincoscos
22
0
00
0
0
1
22222
0
20
22
0
022222
0
2
22
00
cm
cFb
mF
cm
mFa
고유주파수 구동력의 주파수
cycles/sec2
0
cycles/sec
2
School of Mechanical Systems Engineering Engineering Mathematics I
2.8 Modeling: Forced Oscillations. Resonance
주기적인 외력이 가해지는 경우
– 비감쇠 강제진동(c=0). 공진(resonance)
0
ttm
Fy
tm
Fyy
tFkymy
tFtrkycymy
p 0
0
0
002
0
00
0
sin2
cos''
cos''
cos'''
School of Mechanical Systems Engineering Engineering Mathematics I
2.8 Modeling: Forced Oscillations. Resonance
주기적인 외력이 가해지는 경우
– 비감쇠 강제진동(c=0). 맥놀이(beats)
000
ttm
Fy
ttm
Fy
tm
FtCy
2sin
2sin
2
coscos
coscos
00
22
0
0
022
0
0
22
0
00
9.99,1000
School of Mechanical Systems Engineering Engineering Mathematics I
2.8 Modeling: Forced Oscillations. Resonance
주기적인 외력이 가해지는 경우
– 감쇠 강제진동 • 과도해(transient solution): 비제차 방정식의 일반해
• 정상상태해(steady-state solution): 비제차 방정식의 특수해
• 과도해 → 정상상태해
22
0
2
0max
22222
0
2
0
4*
*
cos*sincos
cmc
FC
cm
FC
tCtbtay p
amplitude
10 Fkm
1.0,25.0,5.0,1,2cc
practical resonance
School of Mechanical Systems Engineering Engineering Mathematics I
2.8 Modeling: Forced Oscillations. Resonance
Tacoma Narrows Bridge
School of Mechanical Systems Engineering Engineering Mathematics I
2.9 Modeling: Electric Circuits
전기회로(electric circuit)의 모델화
– 전기회로의 기본 구성요소(RLC 회로) • 저항(resistor), R[Ω]
• 인덕터(inductor), L[H]
• 커패시터(capacitor), C[F]
pipe flow: (pressure difference)~(fluid flow) (voltage difference)~(current, I)
IR
IdtCC
Q 1
dt
dIL
School of Mechanical Systems Engineering Engineering Mathematics I
2.9 Modeling: Electric Circuits
전기회로(electric circuit)의 모델화
– Kirchhoff의 전압 법칙: 폐루프(closed loop)위에 부여된 전압(기전력)은 다른 요소들 양단의 전압 강하의 합
– RLC 회로의 모델화
0
022
22
022
022
0
22
0
0
0
0
,1
tan,,,
sinsincos
cos1
'''
sin1
I
ESRZ
CLS
R
S
b
a
SR
EbaI
SR
REb
SR
SEa
tItbtaI
tEIC
RILI
tEtEdt
dILIdt
CRI
p
reactance impedance
School of Mechanical Systems Engineering Engineering Mathematics I
2.9 Modeling: Electric Circuits
전기량과 역학량의 상사성(analogy)
– 서로 다른 물리적 시스템이 동일한 수학적 모델을 가질 수 있음
– 기계적 시스템에 비하여 전기회로는 구현하기 쉬우며, 물리량의 측정이 빠르고 정확함
전기 시스템 역학(기계적) 시스템
인덕턴스 질량
저항 감쇠계수
커패시턴스의 역수 용수철 상수
기전력의 미분값 구동력
전류 변위
tE cos0 tF cos0
L m
R c
C/1 k
tI ty
School of Mechanical Systems Engineering Engineering Mathematics I
2.10 Solution by Variation of Parameters
매개변수 변환법(method of variation of parameter)
– r(x)가 복잡하여 미정계수법을 이용할 수 없을 경우
– 미분방정식의 변수들(p(x), q(x), r(x))이 연속일 경우에 적용 가능
xryxqyxpy '''
Wronskian:''
0''' ODE, shomogeneou of basis :&
1221
21
12
21
yyyyW
yxqyxpyyy
dxW
ryydx
W
ryyy
yyy
p
ph
School of Mechanical Systems Engineering Engineering Mathematics I
2.10 Solution by Variation of Parameters
매개변수 변환법(method of variation of parameter)
xxCxxCy
xxxxxCxCy
xxxxdxxxdxx
dxxx
xdxxx
x
dxW
ryydx
W
ryyy
xxxxyyyyW
xyxyyy
xxyy
p
sincoscoslnor ,
sincoslncossincos
sincoslncossintancos
1
seccossin
1
secsincos
1sinsincoscos'' Wronskian,
sin,cos0''ODE shomogeneou
cos
1sec''example
21
21
12
21
1221
21
xx coslncos
xxsin
School of Mechanical Systems Engineering Engineering Mathematics I
2.10 Solution by Variation of Parameters
매개변수 변환법(method of variation of parameter)
– 유도 방법
dxW
ryydx
W
ryyvyuyy
dxW
ryv
dxW
ryu
W
ryv
W
ryu
ryvyu
yvyu
ryvyuryvyuqypyyvqypyyu
rvyuyqvyuypvyyvuyyurqypyy
vyyvuyyuyvyuyy
yvyuvyyvuyyuy
xyxvxyxuy
xyCxyCyyyyxryxqyxpy
p
ppp
pp
p
p
h
ph
12
2121
1
2
1
2
21
21
2121222111
21212211
221121
212211
21
2211
'
'
''''
0''
''''''''''''''
'''''''''''''
'''''''''''''
0'''''''
'''
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