Differential Equations
Section 8.2:
Homogeneous Linear Systems with Constant Coefficients
Math 2A
 k1 
 
k
Can we find a solution of the form X   2  et  Ket
 
 
 kn 
for the general homogeneous linear first-order system X’ =AX,
k i are constants, A is an n  n matrix of constants?
Eigenvalues and Eigenvectors:
 k1 
 k1 
 
 
k2  t
k
t

 If X 
e  Ke , where K   2  , is a solution of the linear system, then X '  K  et and
 
 
 
 
 kn 
 kn 
t
t
 X '  AX becomes K  e  AKe
  K  AK
  AK  K  0
 Since K  IK where I is the identity matrix (square matrix with 1’s down the diagonal
and 0’s everywhere else), then ( A   I ) K  0
a1n kn  0 
(a11   )k1  a12 k2  ... 
a k  (a   )k  ... 
a2 n kn  0 
 21 1
21
2
 



an1k1 
an 2 k2  ...  (anm   )kn  0 
  det( A   I )  0 in order for us NOT to have k1  k2  ...  kn  0
 det( A   I )  0 is called the characteristic equation of the matrix A . Its solutions
are eigenvalues of A .
 A solution K  0 corresponding to an eigenvalue  is called an eigenvector of A .
 A solution of the homogeneous system X’ =AX is then
X  Ke t
Three Cases:
Distinct Real Eigenvalues
 1 , 2 ,..., n are distinct real eigenvalues; K1 , K2 ,..., K n are the corresponding eigenvectors
 General solution: X  c1 K1e1t  c2 K 2e2t  ...  cn K n ent
NOTE: The equations can be interpreted as parametric equations of a curve in the xy plane or phase plane.
The curve is called a trajectory. The representative trajectories in the phase plane form a phase portrait.
dx
 2x  2 y
dt
Example:
dy
 x  3y
dt
Repeated Eigenvalues
 There are eigenvalues of multiplicity m
 General solution contains the linear combination c1 K1e1t  c2 K 2e2t  ...  cm K me mt ;
m  n ; K1 , K 2 ,..., K m are linearly independent eigenvectors corresponding to
an eigenvalue 1 of multiplicity m  n
 Only one eigenvector corresponding to the eigenvalue 1 of multiplicity m  n ;
There are m linearly independent solutions of the form
 X 1  K11e1t



1t
1t
 X 2  K 21te  K 22e




 ; K ij are column vectors


t m 1 1t
t m2
1t
1t 
X  K
e  Km2
e  ...  K mn e
m1
 m

(m  1)!
( m  2)!
1 0 0 


Example: X '   0 3 1  X
 0  1 1


Complex Eigenvalues
1     i, 2     i

are complex eigenvalues of A , the coefficient matrix
  0, i 2  1
 A is the coefficient matrix having real entries of the homogeneous system; K1 is the eigenvector
corresponding to the complex eigenvalue 1     i ,  ,  are real; Solution vectors are K1e 1t and K1e1t ;
K1e1t  K1e t (cos(  t )  i sin(  t )

K1e1t  K1et (cos( t )  i sin( t )
Real solutions corresponding to a complex eigenvalue: A is the coefficient matrix of the homogeneous system;
1     i is a complex eigenvalue of A
1


t

 B1  2 ( K1  K1 
 X 1   B1 cos(  t )  B2 sin(  t )  e 

are solutions on  ,  

 Then 
t 
X 2   B2 cos(  t )  B1 sin(  t )  e 


 B  i ( K  K 

2
1
1

2

 dx

 dt  2 x  y  2 z 


 dy

Example 1:   3x  6 z

 dt

 dz

 dt  4 x  3z 


 6  1
Example 2: X '  
 X,
5 4
 2 
X (0)   
 8