Systems of First Order Linear Equations
Ordinary Differential Equations
Dr. Marco A Roque Sol
12/01/2015
Dr. Marco A Roque Sol
Ordinary Differential Equations
Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; Linear Independence, Eig
Basic Theory of Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant Coefficients
Systems of Linear Algebraic Equations; Linear
Independence, Eigenvalues, Eigenvectors
Example 7.7
Solve the system of equations
x1 − 2x2 + 3x3 = b1
−x1 + x2 − 2x3 = b2
2x1 − x2 − 3x3 = b3
for various values of b1 , b2 , and b3
Solution
By performing steps (a), (b), and (c) as in Example 7.6, we
transform the matrix into
Dr. Marco A Roque Sol
Ordinary Differential Equations
Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; Linear Independence, Eig
Basic Theory of Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant Coefficients
Systems of Linear Algebraic Equations; Linear
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
1 −2


0

0
1
0

b1


−1 −b1 − b2 

0 b1 + 3b2 + b3
3
The equation corresponding to the third row is
b1 + 3b2 + b3 = 0
Dr. Marco A Roque Sol
Ordinary Differential Equations
Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; Linear Independence, Eig
Basic Theory of Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant Coefficients
Systems of Linear Algebraic Equations; Linear
Independence, Eigenvalues, Eigenvectors
thus the system has no solution unless the above condition is
satisfied by b1 , b2 , and b3 .
b1 = −3b2 − b3
Assuming that the condition is satisfied


1 −2 3 −3b2 − b3




−(−3b2 − b3 ) − b2 
0 1 −1 

0 0
0 (−3b2 − b3 ) + 3b2 + b3
Dr. Marco A Roque Sol
Ordinary Differential Equations
Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; Linear Independence, Eig
Basic Theory of Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant Coefficients
Systems of Linear Algebraic Equations; Linear
Independence, Eigenvalues, Eigenvectors


1 −2 3 −3b2 − b3




0 1 −1 2b2 + b3 


0 0
0 0
Add (2) times the second row to the first row.


1 0 1 −3b2 − b3




b2 + b3 
0 1 −1 

0 0 0 0
Dr. Marco A Roque Sol
Ordinary Differential Equations
Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; Linear Independence, Eig
Basic Theory of Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant Coefficients
Systems of Linear Algebraic Equations; Linear
Independence, Eigenvalues, Eigenvectors
Thus, we have two equations and one unknown, so one of the
variables let’s say x3 , is equal to a parameter α, obtaining the
system
x1 + α = −3b2 − b3
x2 − α =
b2 + b3
Hence, we obtain
x1 = −α − 3b2 − b3 ;
x2 = α + b2 + b3



 

−α − 3b2 − b3
−1
−3b2 − b3
α + b2 + b3  = α  1  + 
b2 + b3 
X=
α
1
0
Dr. Marco A Roque Sol
Ordinary Differential Equations
Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; Linear Independence, Eig
Basic Theory of Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant Coefficients
Systems of Linear Algebraic Equations; Linear
Independence, Eigenvalues, Eigenvectors
Linear Dependence and Independence .
A set of k vectors x(1) , ..., x(k) is said to be linearly dependent if
there exists a set of real or complex numbers c1 , ...., ck , at least
one of which is nonzero, such that
c1 x(1) + ... + ck x(k) = 0
On the other hand, if the only set c1 , ..., ck for which the above
equation is satisfied is c1 = c2 = · · · = ck = 0,then the set of
vectors x(1) , ..., x(k) is called linearly independent.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; Linear Independence, Eig
Basic Theory of Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant Coefficients
Systems of Linear Algebraic Equations; Linear
Independence, Eigenvalues, Eigenvectors
Consider now a set of n vectors, each of which has n components ,
 
 
 
x1n
x12
x11
x2n 
x22 
x21 
 
 
 
x(1) =  .  ; x(2) =  .  ; · · · x(n) =  . 
 .. 
 .. 
 .. 
xnn
xn2
xn1
the above equation can be written as .


x11 c1 + x12 c2 + . . . + x1n cn
 x21 c1 + x22 c2 + . . . + x2n cn 



=0
..
..


.
.
xn1 c1 + xn2 c2 + . . . + xnn cn
Dr. Marco A Roque Sol
Ordinary Differential Equations
Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; Linear Independence, Eig
Basic Theory of Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant Coefficients
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or equivalently
Xc = 0
(X−1
If X is nonsingular
exists), then the only solution of is c = 0,
−1
but if X is singular (X does not exist) there are nonzero
solutions.
Example 7.8
Determine wether the vectors are linearly indepent or not


 


1
2
− 4
2 ; x(2) =  1 ; x(3) = 
1
x(1) = 
− 1
3
− 11
Dr. Marco A Roque Sol
Ordinary Differential Equations
Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; Linear Independence, Eig
Basic Theory of Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant Coefficients
Systems of Linear Algebraic Equations; Linear
Independence, Eigenvalues, Eigenvectors
Solution
To determine whether x (1) , x (2) , and x (3) are linearly dependent,
we seek constants c1 , c2 , and c3 such that
c1 x(1) + c2 x(2) + c3 x(3) = 0
written in the matrix form

 
1 2
4
c1
2 1
1  c2  = 0
−1 3 −11
c3
Dr. Marco A Roque Sol
Ordinary Differential Equations
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Homogeneous Linear Systems with Constant Coefficients
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Using elementary row operations on the

1 2 −4 

1
2 1

−1 3 −11 augmented matrix

0


0

0
(a) Add (−2) times the first row to the second row, and add the
first row to the third row.


1 2
−4 0




0
−3
9

0


0 5 −15 0
Dr. Marco A Roque Sol
Ordinary Differential Equations
Systems of First Order Linear Equations
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Basic Theory of Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant Coefficients
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(b) Divide the second row by (−3),
second row to the third row.

1 2 −4


0 1 −3

0 0 0
then add (−5) times the

0


0

0
Thus we obtain the equivalent system
c1 + 2c2 − 4c3 = 0
=0
c2 − 3c3 = 0
Dr. Marco A Roque Sol
Ordinary Differential Equations
Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; Linear Independence, Eig
Basic Theory of Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant Coefficients
Systems of Linear Algebraic Equations; Linear
Independence, Eigenvalues, Eigenvectors
Hence, we have 2 equations in 3 unknowns, so one of them, let’s
say c3 will be a free parameter (real number) α and the solution of
the system is
c3 = α;
c2 = 3c3 = 3α;
c1 = −2c2 + 4c3 = −2c3 = −2α



  
c1
− 2α
− 2
c2  = 
3α = α 
3
c3
α
1
Hence, there are infinitely solutions and the set of vectors is
linearly dependent.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; Linear Independence, Eig
Basic Theory of Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant Coefficients
Systems of Linear Algebraic Equations; Linear
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Determinants
Associated to every n × n matrix A there is a real number called
the determinant of A denoted by |A| or det(A) and defined
inductivly as follows
n = 1 A = a11
a11 a12
n=2 A=
a21 a22
|A| = a11
a11 a12 = a11 a22 − a12 a21
|A| = a21 a22 Dr. Marco A Roque Sol
Ordinary Differential Equations
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

a11 a12 a13
n = 3 A = a21 a22 a23 
a31 a32 a33
a11 a12 a13 |A| = a21 a22 a23 =
a31 a32 a33 a22 a23 a21 a23 a11 a12 − a12 a11 a31 a33 + a13 a31 a32 = a11 a22 a33 +
a32 a33 a12 a23 a31 + a13 a21 a32 − a31 a22 a13 − a32 a23 a11 − a33 a21 a12 =
Dr. Marco A Roque Sol
Ordinary Differential Equations
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a11 a12 a13 a11 a12 a11 a22 a33 + a12 a23 a31
a21 a22 a23 a21 a22 +a13 a21 a32 − a31 a22 a13 = a31 a32 a33 a31 a32 −a32 a23 a11 − a33 a21 a12
− − − + + + Now, for n ≥ 4, if let M1j be the corresponding minors to the first
row, then we have
n
X
|A| =
(−1)1+j a1j M1j
j=1
or using any fix row i
|A| =
n
X
j=1
Dr. Marco A Roque
Sol
(−1)i+j aij Mij
Ordinary Differential Equations
Systems of First Order Linear Equations
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Theorem 7.1
Given an n × n matrix A, if B is an n × n matrix obtained from A
by
1) Adding a multiple of the ith row (column) to the jth row then
|B| = |A|
2) Interchanging two consecutive rows (columns), then |B| = −|A|
3) Multiplying a row (column) by a nonzero scalar α then
|B| = α|A|
Dr. Marco A Roque Sol
Ordinary Differential Equations
Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; Linear Independence, Eig
Basic Theory of Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant Coefficients
Systems of Linear Algebraic Equations; Linear
Independence, Eigenvalues, Eigenvectors
Theorem 7.2
1) If A has a row (column) of zeros, then |A| = 0
2) If A has a two identical rows (columns), then |A| = 0
3) If two rows (columns) of A are proportional, then |A| = 0
4) If A ia upper (lower) triangular matrix, then
|A| = a11 a22 a33 · · · ann
Dr. Marco A Roque Sol
Ordinary Differential Equations
Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; Linear Independence, Eig
Basic Theory of Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant Coefficients
Systems of Linear Algebraic Equations; Linear
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5) |AT | = |A|
6) |AB| = |A||B|
Example 7.9
Find the following determinant

1
−1
A=
0
−3
of the matrix
Dr. Marco A Roque Sol

−1 2
4
3 −2 1 

2
1
0
1
1 −1
Ordinary Differential Equations
Systems of First Order Linear Equations
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Basic Theory of Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant Coefficients
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Solution
1 −1 2
4
−1 3 −2 1 =
|A| = 2
1
0 0
−3 1
1 −1
1 −1 2 4 0 2 0 5 0 0 1 −5 =
0 0 7 16 1 −1 2 4 0 2 0 5 0 2 1 0 =
0 −2 7 11
1 −1 2 4 0 2 0 5 0 0 1 −5 = (1)(2)(1)(51) = 102
0 0 0 51 Dr. Marco A Roque Sol
Ordinary Differential Equations
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Theorem 7.3
A matrix A is singular ⇐⇒
|A| = 0 ⇐⇒ Ax = 0 has a
nonzero solution ⇐⇒ Columns of A are linearly dependent.
Eigenvalues and Eigenvectors.
The equation
Ax = y
can be viewed as a linear transformation that maps (or transforms)
a given vector x into a new vector x.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Systems of First Order Linear Equations
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Basic Theory of Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant Coefficients
Systems of Linear Algebraic Equations; Linear
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Given an n × n matrix A we consider the problem of finding a
vector x that is transformed into a multiple of itself
Ax = λx
but this is equivalent to say that
Ax = λIx
Ax − λIx = 0
(A − λI) x = 0
Dr. Marco A Roque Sol
Ordinary Differential Equations
Systems of First Order Linear Equations
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Basic Theory of Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant Coefficients
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The latter equation has nonzero solutions if and only if λ is chosen
so that
|A − λI| = 0
This is a polynomial equation of degree n in λ and is called the
characteristic equation of the matrix A
Values of λ may be either real- or complex-valued and are called
eigenvalues of A . The nonzero vectors that are obtained by using
such a value of λ are called the eigenvectors corresponding to
that eigenvalue.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; Linear Independence, Eig
Basic Theory of Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant Coefficients
Systems of Linear Algebraic Equations; Linear
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a) It is possible to show that if λ1 and λ2 are two eigenvalues of A
and if λ1 6= λ2 , then their corresponding eigenvectors x (1) and x (2)
are linearly independent.
This result extends to any set λ1 , ..., λk of distinct eigenvalues:
their eigenvectors x (1) , ..., x (k) are linearly independent. Thus, if
each eigenvalue of an n × n matrix is simple, then the n
eigenvectors of A , one for each eigenvalue, are linearly
independent.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Systems of First Order Linear Equations
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Basic Theory of Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant Coefficients
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b) On the other hand, if A has one or more repeated eigenvalues,
then there may be fewer than n linearly independent eigenvectors
associated with A, since for a repeated eigenvalue with multiplicity
m, we may have q < m linearly independent vectors.
c) In the case of an eigenvalue, λi with multiplicity m, if we can
find m eigenvectors x (i1) , x (i2) , ..., x (im) , linearly independent
associated to λi , we say that the matrix is Non-defective.
d) Otherwise, if we are able to find just x (i1) , x (i2) , ..., x (iq) ;
q < m linearly independent associated to λi , we say that the
matrix is Defective.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; Linear Independence, Eig
Basic Theory of Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant Coefficients
Systems of Linear Algebraic Equations; Linear
Independence, Eigenvalues, Eigenvectors
Example 7.10
Find the eigenvalues and eigenvectors

0 1
A = 1 0
1 1
of the matrix

1
1
0
Solution
The eigenvalues λ and eigenvectors x satisfy the equation
(A − λI) x = 0, or
Dr. Marco A Roque Sol
Ordinary Differential Equations
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
   
−λ 1
1
x1
0





1 −λ 1
x2 = 0
|A − λI| =
1
1 −λ
x3
0
The eigenvalues are the roots of the equation
−λ 1
1 −λ 1 1 −λ 1 1 1 = 1
1 −λ =
|A − λI| = 1 −λ 1 = − −λ 1
1
1
1 −λ
1 −λ −λ 1
1 1
−λ
1 0
λ+1
−1 − λ = −λ3 + 3λ2 + 2 = 0
0 −λ2 + 1 λ + 1 Dr. Marco A Roque Sol
Ordinary Differential Equations
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The roots of are λ1 = 2, λ2 = −1, and λ3 = −1 .
1) For λ1 = 2

   
−2 1
1
x1
0
 1 −2 1  x2  = 0
1
1 −2
x3
0
We can reduce this to

2
0
0
the equivalent system
   
−1 −1
x1
0




1 −1
x 2 = 0
0
0
x3
0
Dr. Marco A Roque Sol
Ordinary Differential Equations
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by elementary row operations. Solving this system yields the
eigenvector
 
1
(1)

x = 1
1
2) For λ2 = −1

   
1 1 1
x1
0
1 1 1 x2  = 0
1 1 1
x3
0
Dr. Marco A Roque Sol
Ordinary Differential Equations
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The above system is reduced immediately to the single equation
x1 + x2 + x3 = 0
One equation and three unknowns. Hence, two of them are free
veriables, let’s say x1 = c1 , x2 = c2 , and x3 = −c1 − c2 . Thus we
have


 
 
c1
1
0





c2
x=
= c1 0 + c2 1 
−c1 − c2
−1
−1
Dr. Marco A Roque Sol
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In this way two linearly independent eigenvectors associated to
λ2 = −1 are




1
0
0 x(3) = 
1
x(2) = 
− 1
− 1
Thus, the three linearly independent eigenvectors, are
 




1
1
0
0 x(3) = 
1
x(1) = 1 x(2) = 
1
− 1
− 1
Dr. Marco A Roque Sol
Ordinary Differential Equations
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Example 7.11
Find the eigenvalues and eigenvectors of the matrix


2 −3 −1
A =  0 −1 0 
−1 1
2
Solution
The eigenvalues λ and eigenvectors x satisfy the equation
(A − λI) x = 0, or
Dr. Marco A Roque Sol
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
   
2−λ
−3
−1
x1
0





0
−1 − λ
0
x2 = 0
|A − λI| =
−1
1
2−λ
x3
0
The eigenvalues are the roots of the equation
2 − λ
2 − λ
−3
−1 −3
−1 −1 − λ
0 = − −1
1
2 − λ =
|A − λI| = 0
−1
1
2−λ
0
−1 − λ
0 −1
1
2 − λ −1
1
2 − λ 2 − λ
−3
−1 = 0 −1 − λ (2 − λ)2 − 1 =
0
−1 − λ
0 0 −1 − λ
0
Dr. Marco A Roque Sol
Ordinary Differential Equations
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−1
2
−
λ
1
2
− 0 (2 − λ) − 1 −1 − λ = (1 + λ) (2 − λ)2 − 1 = 0
0
0
−1 − λ
The roots of are λ1 = −1, λ2 = 1, and λ3 = 3 .
1) For λ1 = −1


2−λ
−3
−1
−1 − λ
0 =
(A − λ1 I) x =  0
−1
1
2−λ
Dr. Marco A Roque Sol
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
   
3 −3 −1
x1
0
0
0
0  x2  = 0
−1 1
3
x3
0
We can reduce this to the equivalent system

 
 
   
3 −3 −1
1 1
3
1 1 3
x1
0
0 0
0  = 0 0
0  = 0 0 8 x2  = 0
x3
1 1
3
3 −3 −1
0 0 0
0
Dr. Marco A Roque Sol
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Solving this system yields the eigenvector
 
1
x(1) = 1
0
2) For λ2 = 1



2−λ
−3
−1
= 1 −3 −1
−1 − λ
0   0 −2 0  =
(A − λ2 I) x =  0
−1
1
2−λ
−1 1
1
Dr. Marco A Roque Sol
Ordinary Differential Equations
Systems of First Order Linear Equations
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Basic Theory of Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant Coefficients
Systems of Linear Algebraic Equations; Linear
Independence, Eigenvalues, Eigenvectors

 
   
1 −3 −1
1 −3 −1
x1
0
0 −2 0  = 0 −2 0  x2  = 0
0 −2 0
0 0
0
x3
0
Solving this system yields the eigenvector
 
1
(1)

x = 0
1
3) For λ3 = 3


2−λ
−3
−1
−1 − λ
0 =
(A − λ3 I) x =  0
−1
1
2−λ
Dr. Marco A Roque Sol
Ordinary Differential Equations
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Basic Theory of Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant Coefficients
Systems of Linear Algebraic Equations; Linear
Independence, Eigenvalues, Eigenvectors

 

−1 −3 −1
−1 −3 −1
 0 −4 0  =  0 −4 0  =
−1 1
1
0
4
0
   

0
x1
−1 −3 −1
 0 −4 0  x2  = 0
0
x3
0
0
0
Solving this system yields the eigenvector
 
−1
x(1) =  0 
1
Dr. Marco A Roque Sol
Ordinary Differential Equations
Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; Linear Independence, Eig
Basic Theory of Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant Coefficients
Systems of Linear Algebraic Equations; Linear
Independence, Eigenvalues, Eigenvectors
Thus, the three linearly independent eigenvectors, are
x(1)
 
1

= 1
0

x(2)

1
=  0
1
Dr. Marco A Roque Sol

x(3)

− 1
0
=
1
Ordinary Differential Equations
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Example 7.12
Find the eigenvalues and eigenvectors

4 6
A = 1 3
1 −5
of the matrix

6
2
−2
Solution
The eigenvalues λ and eigenvectors x satisfy the equation
(A − λI) x = 0, or
Dr. Marco A Roque Sol
Ordinary Differential Equations
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Basic Theory of Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant Coefficients
Systems of Linear Algebraic Equations; Linear
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
   
4−λ
6
6
x1
0





1
3−λ
2
x2 = 0
|A − λI| =
−1
−5 −2 − λ
x3
0
The eigenvalues are the roots of the equation
4 − λ
4 − λ
6
6
6
6
3−λ
2 = 1
3−λ
2 =
|A − λI| = 1
−1
−5 −2 − λ 0
−2 − λ −λ
1
3
−
λ
2
− 0 −(4 − λ)(3 − λ) + 6 6 − 2(4 − λ) = −λ3 + 5λ2 − 8λ + 4 =
0
−2 − λ
−λ
Dr. Marco A Roque Sol
−(λ − 1)(λ − 2)2 = 0
Ordinary Differential Equations
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The roots of are λ1 =, λ2 = 2, and λ3 = 2 .
1) For λ1 = 1
 
  
3
6
6
x1
4−λ
6
6
x1







x2 
1
2
2
1
3−λ
2
x2 =
(A − λ1 I) x =
x3
x3
−1 −5 −3
−1
−5 −2 − λ

We can reduce this to the equivalent system

  
  
   
3 6 6
x1
1 2 2
x1
1 2
2
x1
0
1 2 2 x2  = 1 2 2 x2  = 1 −3 −1 x2  = 0
0 3 1
x3
0 3 1
x3
0 0
0
x3
0
Dr. Marco A Roque Sol
Ordinary Differential Equations
Systems of First Order Linear Equations
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Basic Theory of Systems of First Order Linear Equations
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Solving this system yields the eigenvector
 
4
x(1) =  1 
−3
2) For λ2 = 2

 
  
4−λ
6
6
2
6
6
x1







1
3−λ
2
1
1
2
x2 
(A − λ2,3 I) x =
=
−1
−5 −2 − λ
−1 −5 −4
x3

 
 
    
2
6
6
1 1
2
1 1 2
x1
0
 1









1
2
2
x 2 = 0
= 0 4
= 0 4 2
−1 −5 −4Dr. Marco A0Roque
−4
0 0Equations
0
x3
0
Sol −2
Ordinary Differential
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The above system is reduced immediately to the equations
x1 + x2 + 2x3 = 0
4x2 + 2x3 = 0
Two equations and three unknowns. Hence, one of them 1 is free
veriables, let’s say x3 = α, x2 = 21 α, and x3 = −2x3 − x2 = −3α .
Thus we have




− 3α
− 3
1 
1
x=
= α
2α
2
− 3α
− 3
Dr. Marco A Roque Sol
Ordinary Differential Equations
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In this way, there is one linearly independent eigenvector associated
to λ2,3 = 2 , namely,


3
1
x(2) = 
− 2
Therefore, there are just two linearly independent eigenvectors
 


1
4
1
x(1) = 1 x(2) = 
1
− 3
that is, the matrix A is defective
Dr. Marco A Roque Sol
Ordinary Differential Equations
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Basic Theory of Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant Coefficients
Systems of Linear Algebraic Equations; Linear
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Example 7.13
Find the eigenvalues and eigenvectors of the matrix


1 0 0
A = 2 1 −2
3 2 1
Solution
The eigenvalues λ and eigenvectors x satisfy the equation
(A − λI) x = 0, or
Dr. Marco A Roque Sol
Ordinary Differential Equations
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Homogeneous Linear Systems with Constant Coefficients
Systems of Linear Algebraic Equations; Linear
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
   
1−λ
0
0
x1
0





2
1 − λ −2
x2 = 0
(A − λI) x =
3
2
1−λ
x3
0
The eigenvalues are the roots of the equation
1 − λ
0
0
1 − λ −2 =
1 − λ −2 = (1 − λ) |A − λI| = 2
2
1
−
λ
3
2
1 − λ
(1 − λ)3 + 4(1 − λ) = (1 − λ)(λ2 − 2λ + 5) = 0
Dr. Marco A Roque Sol
Ordinary Differential Equations
Systems of First Order Linear Equations
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Homogeneous Linear Systems with Constant Coefficients
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Independence, Eigenvalues, Eigenvectors
The roots of are λ1 = 1, λ2 = 1 + 2 i, and λ3 = 1 − 2 i .
1) For λ1 = 1

 
    
0
1−λ
0
0
0 0 0
x1









2
1 − λ −2
x2 = 0
(A − λ1 I) x =
= 2 0 −2
x3
0
3
2
1−λ
3 2 0
We can reduce this to the equivalent system

 
 
    
2 0 −2
1 0 −1
1 0 −1
x1
0
0 0 0  = 0 0 0  = 0 2 4  x2  = 0
3 2 0
3 2 0
0 0 0
x3
0
Dr. Marco A Roque Sol
Ordinary Differential Equations
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Solving this system yields the eigenvector


1
x(1) = − 3/2
1
2) For λ2 = 1 + 2 i

 
1−λ
0
0
x1
1 − λ −2  x2  =
(A − λ2 I) x =  2
3
2
1−λ
x3

   
−2 i
0
0
x1
0
 2




−2 i −2
x2 = 0
3
2
−2 i
x3
0
Dr. Marco A Roque Sol
Ordinary Differential Equations
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Independence, Eigenvalues, Eigenvectors
The above system is reduced immediately to the equations
x1 = 0
− 2 ix2 − 2x3 = 0,
2x2 − 2 ix3 = 0
Thus, we have one equation and two unknowns. Hence, one of
them 1 is a free veriable, let’s say x2 = α, x3 = −i α . Hence




 
 
0
0
0
0








α
1 =α
1 −i
0
=α
x=
−i α
− i
0
1
Dr. Marco A Roque Sol
Ordinary Differential Equations
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In this way, there are two real linearly independent eigenvector
associated to λ2 = 1 + 2 i , namely,
 
 
0
0
(2)
(3)



1
0
x =
x =
0
1
3) For λ3 = 1 − 2 i

 
1−λ
0
0
x1



2
1 − λ −2
x2  =
(A − λ2 I) x =
3
2
1−λ
x3

   
2i 0
0
x1
0
 2 2 i −2 x2  = 0
3 A2Roque2Soli
x3 Differential
0 Equations
Dr. Marco
Ordinary
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Homogeneous Linear Systems with Constant Coefficients
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The above system is reduced immediately to the equations
x1 = 0;
2i x2 − 2x3 = 0;
2x2 + 2i x3 = 0
Thus, we have one equation and two unknowns. Hence, one of
them 1 is free variable, let’s say x2 = α, x3 = i α . Thus we have
 
 
 
 
0
0
0
0








1 =α
1 +i
0
x= α =α
iα
i
0
1
Dr. Marco A Roque Sol
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In this way, there is two real linearly independent eigenvector
associated to λ2 = 1 + 2 i , namely,
 
 
0
0
(2)
(3)



1
0
x =
x =
0
1
Hence, we have three linearly independent eigenvectors, namely


 
 
1
0
0
x(1) = − 3/2 x(2) = 1 x(3) = 0
1
0
1
that is, the matrix A is Non-defective
Dr. Marco A Roque Sol
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OBS
Let A be a real-valued n × n matrix. If x(1) = x(R) ± i x(I ) are
complex eigenvectors of the matrix A with complex eigenvalues
λ = u ± i v . then, x(R) and x(I ) are two real eigenvectors for A
with eigenvalues λ = u ± i v
Finally, let’s introduce another concept from linear algebra
The Dot Product
For y = (y1 , y2 , ..., yn ), x = (x1 , x2 , ..., xn ) ∈ R, define the dot
product or inner product or scalar product.as
Dr. Marco A Roque Sol
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x · y =< x, y >= x1 x2
 
y1


y2 
. . . xn  .  = x1 y1 + x2 y2 + ... + xn yn
 .. 
yn
OBS
a) x and y are said to be orthogonal if < x, y >= 0 .
b) Orthogonal nonzero vectors ae linearly independent.
Dr. Marco A Roque Sol
Ordinary Differential Equations
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Theorem 7.10
Let A be an n × n matrix. If A is symetric, ( A = AT ) then
1) All eigenvalues are real.
2) A is always Nondefective.
3) The eigenvectors corresponding to different eigenvalues are
orthogonal, thus if λ1 , λ2 , ..., λn are all simple, v1 , v2 , ..., vn form
an orthogonal set.
Dr. Marco A Roque Sol
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Dr. Marco A Roque Sol
Ordinary Differential Equations
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Basic Theory of Systems of First Order Linear Equations
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Equations
The general theory of a system of n first order linear equations
x10 = p11 x1 + p12 x2 + . . . + p1n xn + g1 (t)
x20 = p21 x1 + p22 x2 + . . . + p2n xn + g2 (t)
..
..
.
.
xn0 = pn1 x1 + pn2 x2 + . . . + pnn xn + gn (t)
or
X0 = P(t)X + g(t)
closely parallels that of a single linear equation of nth order.
Dr. Marco A Roque Sol
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we assume that P and g are continuous on some interval
α < t < β; that is, each of the scalar functions
p11 , ..., pnn , g1 , ..., gn is continuous there.
Theorem 7.4
If the vector functions x(1) and x(2) are solutions of the
homogeneus system ( g(t) = 0 ) then the linear combination
c1 x(1) + c2 x(2) is also a solution for any constants c1 and c2 .
This is the principle of superposition
Dr. Marco A Roque Sol
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By repeated application of Theorem, we can conclude that if
x(1) , ..., x(k) are solutions of the homogeneous system, then
c1 x(1) + ... + ck x(k)
is also a solution for any constants c1 , ..., ck .
Theorem 7.5
If the vector functions x(1) , ..., x(n) are linearly independent
solutions of the homogeneous system for each point in the interval
α < t < β, then each solution x = phi(t) of the homogeneous
system can be expressed as a linear combination of x(1) , ..., x(n) in
exactly one way.
Dr. Marco A Roque Sol
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If the constants c1 , ..., cn are thought of as arbitrary, then the
above equation includes all solutions of the system, and it is
customary to call it the general solution.
Any set of solutions x(1) , ..., x(n) of the homogeneus system that is
linearly independent at each point in the interval α < t < β is said
to be a fundamental set of solutions for that interval.
Theorem 7.6
If x(1) , ..., x(n) are solutions of the homogeneus system on the
interval α < t < β, then in this interval W [x(1) , ..., x(n) ] either is
identically zero or else never vanishes.
Dr. Marco A Roque Sol
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To prove this theorem is necessary to establish that
dW
= [p11 + p22 + ... + pnn ]W
dt
Hence
W (t) = ce
R
[p11 +p22 +...+pnn ]dt
Theorem 7.7
Let x(1) , ..., x(n) be the solutions of the homogeneus system that
satisfy the initial conditions x(1) (t0 ) = e(1) , x(1) (t0 ) = e(2) ,
..., x(n) (t0 ) = e(n) , respectively, where t0 is any point inα < t < β
and
Dr. Marco A Roque Sol
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e(1)
 
1
0
 
= .
 .. 
e(2)
 
0
1
 
= .
 .. 
0
···
e(n)
 
0
0
 
= .
 .. 
1
0
Then, x(1) , ..., x(n) form a fundamental set of solutions of the
homogeneous system.
Finally in the case that the solution is complex-valued, we have the
following result.
Dr. Marco A Roque Sol
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Theorem 7.8
Consider the homogeneous system
X0 = P(t)X
where each element of P is a real-valued continuous function. If
x = u(t) + i v(t) is a complex-valued solution, then its real part
u(t) and its imaginary part v(t) are also solutions of this equation.
Dr. Marco A Roque Sol
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Coefficients
We will concentrate most of our attention on systems of
homogeneous linear equations with constant coefficients
x0 = Ax
where A is a constant n × n matrix. Unless stated otherwise, we
will assume further that all the elements of A are real (rather than
complex) numbers.
The case n = 2 is particularly important and lends itself to
visualization in the x1x2 − plane, called the phase plane. By
evaluating Ax at a large number of points and plotting the
resulting vectors, we obtain a direction field of tangent vectors to
solutions of the system of differential equations.
Dr. Marco A Roque Sol
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Coefficients
A qualitative understanding of the behavior of solutions can usually
be gained from a direction field. More precise information results
from including in the plot some solution curves, or trajectories. A
plot that shows a representative sample of trajectories for a given
system is called a phase portrait .
Dr. Marco A Roque Sol
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Coefficients
Now, for the system
x0 = Ax
we look for solutions of the form
x = ve λt
where the expon entλ and the vector v are to be determined.
Substituting x in the system gives
λve λt = Ave λt
(A − λI) v = 0
Dr. Marco A Roque Sol
Ordinary Differential Equations
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Coefficients
Thus, to solve the system of differential equations, we must solve
the above system of algebraic equations. That is, we need to find
the eigenvalues and eigenvectors of the matrix A.
If we assume that A is a real-valued matrix, then we must consider
the following possibilities for the eigenvalues of A:
1. All eigenvalues are real and different from each other.
2. Some eigenvalues occur in complex conjugate pairs.
3. Some eigenvalues, either real or complex, are repeated. If the n
eigenvalues are all real and different, as in the
Dr. Marco A Roque Sol
Ordinary Differential Equations
Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; Linear Independence, Eig
Basic Theory of Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant Coefficients
Homogeneous Linear Systems with Constant
Coefficients
Example 7.14
Consider the system
1 1
x = Ax
x
4 1
0
Solution
Let’s find the eigenvalues of the matrix A
Dr. Marco A Roque Sol
Ordinary Differential Equations
Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; Linear Independence, Eig
Basic Theory of Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant Coefficients
Homogeneous Linear Systems with Constant
Coefficients
1 − λ
1 =0
|A − λI| = 4
1 − λ
(1 − λ)2 − 4 = 0 =⇒
(λ2 − 2λ − 3 = (λ − 3)(λ + 1) = 0 =⇒
Dr. Marco A Roque Sol
Ordinary Differential Equations
Systems of Linear Algebraic Equations; Linear Independence, Eig
Basic Theory of Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant Coefficients
Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant
Coefficients
λ1 = 3,
λ2 = −1
If λ1 = 3, then the system reduces to the single equation
−2v1 + v2 = 0, =⇒ v2 = 2v1
and a corresponding eigenvector is
v
(1)
Dr. Marco A Roque Sol
1
=
2
Ordinary Differential Equations
Systems of First Order Linear Equations
Systems of Linear Algebraic Equations; Linear Independence, Eig
Basic Theory of Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant Coefficients
Homogeneous Linear Systems with Constant
Coefficients
Similarly, corresponding to λ2 = −1, we find that a corresponding
eigenvector is
1
(2)
v =
− 2
The corresponding solutions of the differential
1 3t
(1)
(2)
x =
e ; x =
2
−
Dr. Marco A Roque Sol
equation are
1 −t
e
2
Ordinary Differential Equations
Systems of Linear Algebraic Equations; Linear Independence, Eig
Basic Theory of Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant Coefficients
Systems of First Order Linear Equations
Homogeneous Linear Systems with Constant
Coefficients
The Wronskian of these solutions is
3t
e
e −t (1) (2)
= −4e 2t 6= 0
W [x , x ](t) = 3t
2e
−2e −t Hence the solutions x(1) and x(2) form a fundamental set, and the
general solution of the system is
x=
c1 x
(1)
+ c2 x
(2)
=
c1
Dr. Marco A Roque Sol
1 3t
e +
2
c2
1 −t
e
− 2
Ordinary Differential Equations