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Engineering Mathematics I- Chapter 4. Systems of ODEs
민기복민기복
Ki-Bok Min PhDKi Bok Min, PhD서울대학교 에너지자원공학과 조교수Assistant Professor, Energy Resources Engineering, gy g g
Ch.4 Systems of ODEs. Phase Plane. Q lit ti M th dQualitative Methods
• Basics of Matrices and Vectors• Systems of ODEs as Models• Systems of ODEs as Models• Basic Theory of Systems of ODEs• Constant-Coefficient Systems. Phase Plane Method
C it i f C iti l P i t St bilit• Criteria for Critical Points. Stability• Qualitative Methods for Nonlinear SystemsQ y• Nonhomogeneous Linear Systems of ODEs
Basics of Matrices and Vectors
• Systems of Differential Equations– more than one dependent variable and more than one equationmore than one dependent variable and more than one equation
1 11 1 12 2' ,y a y a y 1 11 1 12 2 1
2 21 1 22 2 2
' ,' ,
n ny a y a y a yy a y a y a y
2 21 1 22 2' ,y a y a y 2 21 1 22 2 2
1 1 2 2
,
' ,
n n
n n n nn n
y a y a y a y
y a y a y a y
…
• Differentiation– The derivative of a matrix with variable entries is obtained by
differentiating each entry.
1 1
2 2
' '
'y t y t
t ty t y t
y y
Basics of Matrices and VectorsEivenvalues (고유치) and Eivenvectors (고유벡터)Eivenvalues (고유치) and Eivenvectors (고유벡터)
• Let be an n x n matrix. Consider the equation where is a scalar and x is a vector to be determined.
jka A Ax x
– A scalar such that the equation holds for some vector x ≠ 0 is called an eigenvalue of A,
Ax x
– And this vector is called an eigenvector of A corresponding to this eigenvalue .
A Ax x
Basics of Matrices and VectorsEivenvalues (고유치) and Eivenvectors (고유벡터)Eivenvalues (고유치) and Eivenvectors (고유벡터)
Ax x 0 A I x0 Ax Ix
– n linear algebraic equations in the n unknowns (the components of x).
1, , nx x
p )– The determinant of the coefficient matrix A I must be zero in
order to have a solution x ≠ 0.
• Characteristic Equation
– Determine λ1 and λ2
det 0 A IDetermine λ1 and λ2
– Determination of eigenvector corresponding to λ1 and λ2
Basics of Matrices and VectorsEivenvalues (고유치) and Eivenvectors (고유벡터)Eivenvalues (고유치) and Eivenvectors (고유벡터)
• Example 1. Find the eigenvalues and eigenvectors
0404
2.16.10.40.4
A
• If x is an eigenvector, so is kxg ,
Systems of ODEs as Models
10 gal/min
10 gal/min
10 gal/min
10 gal/minVS.
10 gal/min
Single ODE Systems of ODEs
Systems of ODEs as ModelsE l 3 Mi i bl (S 1 3)Example 3. Mixing problem (Sec 1.3)
– Initial Condition: 1000 gal of water, 100 lb salt, initially brine runs in 10 gal/min, 5 lb/gal, stirring all the time, brine runs out at 10 gal/mingal/min
– Amount of salt at t?
10 gal/min
10 gal/min
?
Systems of ODEs as Models
– Tank T1 and T2 contain initially 100 gal of water each.
– In T1 the water is pure, whereas 150 lb of fertilizer are dissolved in T2.
– By circulating liquid at a rate of 2 gal/min and stirring the amounts of fertilizer
in T1 and in T2 change with time t. 2y t
1y t
1 2 g
– Model Set Up2 gal/min
2y
2 gal/min
1 2 1 1 1 1 22 2' Inflow / min - Outflow / min Tank ' 0.02 0.02
100 100y y y T y y y
2 1 2 2 2 1 22 2' Inflow / min - Outflow / min Tank ' 0.02 0.02
100 100y y y T y y y
0 02 0 02 0.02 0.02 ' ,
0.02 0.02
y Ay A
Systems of ODEs as Models
y1-y2 plane - phase plane (상평면 )Trajectory (궤적) : -상평면에서의곡선-전체해집합의일반적인양상을잘표현함.Phase portrait (상투영): trajectories in phase plane
Systems of ODEs as Models
• Example 2. Electrical Network
y1-y2 plane - phase plane (상평면 )Trajectory (궤적) : -상평면에서의곡선-전체해집합의일반적인양상을잘표현함.Phase portrait (상투영): trajectories in phase plane
Systems of ODEs as ModelsCon ersion of an nth order ODE to a s stemConversion of an nth-order ODE to a system
• 2nd order to a system of ODE – Example 3. Mass on a Spring
0''' k 0''' kycymy
21' yy 10 y' , 21 yyyy
2
1
yy
y
212
21
' ymcy
mky
yy
2
10
'yy
mc
mkAyy
01
det 2mk
mcck
IA Calculation same
as before!mmmm as before!
) '' 2 ' 0 75 0E ) '' 2 ' 0.75 0Ex y y y
Systems of ODEs as ModelsCon ersion of an nth order ODE to a s stemConversion of an nth-order ODE to a system
– solve single ODE by methods for systemsg y y– includes higher order ODEs into …first order ODE
Basic Theory of Systems of ODEsC t ( l t i l ODE)Concepts (analogy to single ODE)
• First-Order Systems 1 1 1' , , , ny f t y y
1 1
,
n n
y f
y f
y f
2 2 1
1
' , , ,
' , , ,
n
n n n
y f t y y
y f t y y
n ny f ' ,ty f y
S l ti i t l t b
1n n n
• Solution on some interval a
Basic Theory of Systems of ODEsE i t d U iExistence and Uniqueness
• Theorem 1. Existence and Uniqueness Theorem
– Let be continuous functions having continuous partial
derivatives in some domain R of
1, , nf f
f f f tderivatives in some domain R of -
space containing the point . Then the first-order system
1 1
1, , , ,
n n
f f fy y y
n 1 2 nty y y
0 1, , , nt K Kspace containing the point . Then the first order system
has a solution on some interval satisfying the
0 1 n
0 0t t t
initial condition , and this solution is unique.
Basic Theory of Systems of ODEsH N hHomogeneous vs. Nonhomogeneous
• Linear Systems 'y a t y a t y g t
11 1 1 1
1 2
, , n
n nn n
a a y g
a a y g
A y g
1 11 1 1 1
1 1'
n n
n n nn n n
y a t y a t y g t
y a t y a t y g t
' y Ay g
Homogeneous:– Homogeneous:– Nonhomogeneous: ' , y Ay g g 0
' y Ay
Basic Theory of Systems of ODEsC t ( l t i l ODE)Concepts (analogy to single ODE)
• Theorem 2 (special case of Theorem 1)
Th 3• Theorem 3
(1) (2) (1) (2) (1) (2)1 2 1 2 1 2
(1) (2)1 2
c c c c c c
c c
y y y y y Ay Ay
A y y Ay
Basic Theory of Systems of ODEsG l l tiGeneral solution
미분아님!!• Basis, General solution, Wronskian– Basis: A linearly independent set of n solutions of the 1 , , ny y
미분아님!!
Basis: A linearly independent set of n solutions of the homogeneous system on that interval
– General Solution: A corresponding linear combination
, ,y y
General Solution: A corresponding linear combination 11 1 , , arbitraryn nc c c c ny y y
– Fundamental Matrix : An n x n matrix whose columns are nsolutions 1 , , ny y 1 , ,
nY y y y = Yc– Wronskian of : The determinant of Y 1 , , ny y
1 21 1 1
1 2
n
n
y y y
y y y
ex)
1 2 2 2
1 2
W , ,
nn n n
y y y
y y y
ny y
ex)(1) (2) 0.5 1.5
1 1 0.5 1.51 11 2(1) (2) 0.5 1.5
2 22 2
2 121 1.51.5
t tt t
t t
c cy y e eYc c e c e
c cy y e e
y
Constant Coefficient Systems.
• Homogeneous linear system with constant coefficients
' y Ay
• Homogeneous linear system with constant coefficients – Where n x n matrix has entries not depending on tjka A
– Try tey x
' t te e y x Ay Ax Ax x
• Theorem 1. General Solution
y y
– The constant matrix A in the homogeneous linear system has a linearly independent set of n eigenvectors, then the corresponding solutions form a basis of solutions and the 1 ny ysolutions form a basis of solutions, and the corresponding general solution is
, , y y 11
1nn tt
nc e c e y x x
Constant Coefficient Systems.
• Example 1. Improper node (비고유마디점)
3 1 3 1'1 3
y Ay y 1 1 2
2 1 2
3
3
y y y
y y y
tt ececccy 42211 11
yyy ececccy 21212 11
yyy
Constant Coefficient Systems.
• Example 2. Proper node (고유마디점)1 0
'
y Ay y 1 1y y
0 1 y Ay y
2 2y y
t
ttt
ecyecy
ecec22
1121 1
001
y
Constant Coefficient Systems.
• Example 3. Saddle point (안장점)1 0
'
y Ay y 1 1y y
0 1 y Ay y
2 2y y
ttt ecyecec
1101
y tecyecec
22
21 10y
Constant Coefficient Systems.
• Example 4. Center (중심)0 1
'
y Ay y 1 2y y
4 0 y Ay y
2 14y y
itit
itititit
eiceicyececye
ice
ic 2
22
12
22
2112
22
1 22
21
21
y
Constant Coefficient Systems.
• Example 5. Spiral Point (나선점)1 1 1 1 2y y y 1 1
'1 1
y Ay y2 1 2y y y
titi ei
cei
c
12
11
11y
Constant Coefficient Systems.
• Example 6. Degenerate Node (퇴화마디점): – 고유벡터가기저를형성하지않는경우: 행렬이고유벡터가기저를형성하지않는경우: 행렬이대칭이거나반대칭(skew-symmetric)인경우퇴화마디점은생길수가없음.
texy 1
yy
2114
' exy
tt ete uxy 2
tt etcec 33011
y etcec 21 111
y
Constant Coefficient Systems.SSummary Δ is discriminant (판별식) of
determinant of (A-λI)• Improper node (비고유마디점):
– Real and distinct eigenvalues of the same sign1 20, 0
• Proper node (고유마디점): – Real and equal eigenvalues
*1 20,
q g
• Saddle point (안장점): Real eigenvalues of opposite sign
1 20, 0 – Real eigenvalues of opposite sign
• Center (중심): 0, pure imaginary – Pure imaginary eigenvalues
• Spiral point (나선점): 0, complex p p ( )– Complex conjugates eigenvalues with nonzero real part
* : when two linearly independent eigenvectors exist. Otherwise, degenerate node
Name Δ Eigenvalue Trajectories
Improper node(비고유마디점) 0 1 2 0
Proper node(고유마디점) 0 (고유마디점) 0 1 2
Saddle Point(안장점) 0 1 2 0
Center(중심)
Pure imaginarye.g., i( ) g
Spiral Point Complex number
0
i
Spiral Point(나선점)
Complex numbere.g., 1 i
Criteria for Critical Points. Stability
• Critical Points of the system2
2 2 21 1 22 2''dy
dy y a y a ydt y Ay 2 2 21 1 22 211 1 11 1 12 2
' '
y y y ydtdydy y a y a ydt
y Ay
– A unique tangent direction of the trajectory passing through except for the point
2
1
dydy
0 0P P Pthrough , except for the point
– Critical Points: The point at which becomes undetermined, 0/0
0 : 0,0P P 1 2: ,P y y2
1
dydy1y
Criteria for Critical Points. Stability
• Five Types of Critical Points
– Depending on the geometric shape of the trajectories near them
Improper Node
Proper Node
SSaddle Point
Center
Spiral Point
Criteria for Critical Points. Stability
• Stable Critical point (안정적 임계점):
– 어떤 순간 에서 임계점에 아주 가깝게 접근한 모든궤적이 이후의 시간에서도 임계점에 아주 가까이 접근한
0tt 궤적이 이후의 시간에서도 임계점에 아주 가까이 접근한상태로 남아 있는 경우.
• Unstable Critical point (불안정적 임계점):
– 안정적이 아닌 임계점
• Stable and Attractive Critical point (안정적 흡인임계점)임계점):
– 안정적 임계점이고, 임계점 근처 원판 내부의 한 점을지나는 모든 궤적이 를 취할 때 임계점에 가까이t접근하는 경우
Criteria for Critical Points. Stability
St bl & U t blStable & attractive
Unstable Unstable
Stable Stable & Attractive
Qualitative Methods for Nonlinear S tSystems
• Qualitative Method– Method of obtaining qualitative information on solutions without Method of obtaining qualitative information on solutions without
actually solving a system.– particularly valuable for systems whose solution by analytic particularly valuable for systems whose solution by analytic
methods is difficult or impossible.
N li
1 1 1 2
Nonlinear systems' ,
, thus '
y f y y'
f
y f y 2 2 1 2' ,y f y yy y
'y a y a y h y y linearization 1 11 1 12 2 1 1 2
2 21 1 22 2 2 1 2
,, thus
' ,y a y a y h y yy a y a y h y y
y' Ay h y
linearization
Qualitative Methods for Nonlinear S tSystems
• Theorem 1. Linearization– If f1 and f2 are continuous and have continuous partial derivatives If f1 and f2 are continuous and have continuous partial derivatives
in a neighborhood of the critical point (0,0), and if , then the kind and stability of the critical point of nonlinear systems are th th f th li i d t
det 0A
the same as those of the linearized system
1 11 1 12 2'thy a y a y
' A
– Exceptions occur if A has equal or pure imaginary eigenvalues;
1 11 1 12 2
2 21 1 22 2
, thus ' .y y yy a y a y
y' Ay
Exceptions occur if A has equal or pure imaginary eigenvalues; then the nonlinear system may have the same kind of critical points as linearized system or a spiral point.
Qualitative Methods for Nonlinear S tSystems
• Example 1. Free undamped pendulum– Determine the locations and types of critical pointsDetermine the locations and types of critical points– Step 1: setting up the mathematical model
q : the angular displacement measured counterclockwise q : the angular displacement, measured counterclockwise from the equilibrium position
The weight of the bob : mgThe weight of the bob : mg
A restoring force tangent to the curve of motion of the bob : mg sin q
By Newton’s second law, at each instant this force is balanced by the force of acceleration mLq``
'' sin 0 sin 0 gmLθ mg θ'' k kL
Qualitative Methods for Nonlinear S tSystems
– Step 2: Critical Points Linearization 0,0 , 2 ,0 , 4 ,0 ,
sin 0θ'' k 1 2set , 'y y 1 2'' iy yk
2 1' siny k y
2 10, sin 0 infinitely many critical points 0 , 0, 1, 2, y y nπ , n : 1 '0 1
– Step 3: Critical Points Linearization.3
1 1 1 11sin6
y y y y 1 22 1
'0 1, thus
'0y y
'y kyk
y Ay y
,0 , 3 ,0 , 5 ,0 ,
Consider the critical point (p , 0) 1 2set , ' 'y y 0 1' y Ay y
critical points are all saddle points.
sin 0θ'' k
31 1 1 1 11sin sin sin6
y y y y y 0'
k
y Ay y
Qualitative Methods for Nonlinear S tSystems
Nonhomogeneous Linear Systems of ODEODEs
• Nonhomogeneous of Linear Systems:– Assume g(t) and the entries of the n x n matrix A(t) to be
' , y Ay g g 0Assume g(t) and the entries of the n x n matrix A(t) to be continuous on some interval J of the t-axis.
– General solution : h p y y yGeneral solution : : A general solution of the homogeneous system on J
: A particular solution(containing no arbitrary constants) of
y y y hy py ' y Ay
' y Ay g
: A particular solution(containing no arbitrary constants) of on J
• Methods for obtaining particular solutions
y y Ay
• Methods for obtaining particular solutions– Method of Undetermined Coefficients– Method of the Variation of Parameter
Nonhomogeneous Linear Systems of ODEODEs
• Method of undetermined coefficients: – Components of g : Components of g : – (1) constants
(2) iti i t f t– (2) positive integer powers of t
– (3) exponential functions – (4) cosines and sines.
Nonhomogeneous Linear Systems of ODEODEs
• Example 1. Method of undetermined coefficients. Modification rule 3 1 6
A general solution of the homogeneous system :
23 1 6'1 3 2
ty e y Ay g
2 41 2
1 1h t tc e c e
y– A general solution of the homogeneous system :
– Apply the Modification Rule by setting
Equating the terms on both sides:
1 21 1 y
2 2p t tte e y u v
1 2tt – Equating the -terms on both sides:– Equating the other terms:
6
2 2 h 0k
k A
1
2 with any 01
a a
u Au u2tte
– General solution :
2 2, choose 02 4
a kk
u v Av v
1 1 1 2 2 4 2 21 2
1 1 1 22
1 1 1 2t t t tc e c e te e
y
Nonhomogeneous Linear Systems of ODEODEs
• Example 1. solution by the method of variation of parameters3 1 6
– General solution of the homogeneous system :
23 1 6'1 3 2
ty e y Ay g
General solution of the homogeneous system :
– Particular solution :
11 'h nnc c t y y y Y c Y AY p t ty Y u– Particular solution : t ty Y u
' ' ' ' ' pt
y Y u Yu Y u Yu AYu g
0
1 1 ' ' t
t
t t dt Yu g u Y g u Y g
Nonhomogeneous Linear Systems of ODEODEs
• Example 1. solution by the method of variation of parameters (cont.)
4 4 2 21 1 1
t t t te e e e Y 6 2 2 4 4
2 2 21
2 24 4 2
2 2
4 261 1'8 42 22
t t t t t
t t t
t tt t t
e e e e e
e e ee e
Y
u Y g 4 4 2
2 20
8 42 22
2 24 2 2
t t t
t
t t
e ee e e
tdt
e e
u
0
2 4
2 4
t tp
t t
e ee e
y Yu
2 2 42 4
2 2 2 4
2 2 2 22 2 22 2 2 2 22 2 2
t t tt t
t t t t
t tte e ee e
e tte e e
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