Ch 7.4: Basic Theory of Systems of First
Order Linear Equations
The general theory of a system of n first order linear equations
x1  p11 (t ) x1  p12 (t ) x2    p1n (t ) xn  g1 (t )
x2  p21 (t ) x1  p22 (t ) x2    p2 n (t ) xn  g 2 (t )

xn  pn1 (t ) x1  pn 2 (t ) x2    pnn (t ) xn  g n (t )
parallels that of a single nth order linear equation.
This system can be written as x' = P(t)x + g(t), where
 x1 (t ) 
 g1 (t ) 
 p11 (t )





 x2 (t ) 
 g 2 (t ) 
 p21 (t )
x(t )  
, g(t )  
, P(t )  










 x (t ) 
 g (t ) 
 p (t )
 n 
 n 
 n1
p12 (t )  p1n (t ) 

p22 (t )  p2 n (t ) 


 

pn 2 (t )  pnn (t ) 
Vector Solutions of an ODE System
A vector x = (t) is a solution of x' = P(t)x + g(t) if the
components of x,
x1  1 (t ), x2  2 (t ), , xn  n (t ),
satisfy the system of equations on I:  < t < .
For comparison, recall that x' = P(t)x + g(t) represents our
system of equations
x1  p11 (t ) x1  p12 (t ) x2    p1n (t ) xn  g1 (t )
x2  p21 (t ) x1  p22 (t ) x2    p2 n (t ) xn  g 2 (t )

xn  pn1 (t ) x1  pn 2 (t ) x2    pnn (t ) xn  g n (t )
Assuming P and g continuous on I, such a solution exists by
Theorem 7.1.2.
Example 1
Consider the homogeneous equation x' = P(t)x below, with
the solutions x as indicated.
 1 1
x ;
x  
 4 1
 e3t   1 3t
x(t )   3t    e
 2e   2 
To see that x is a solution, substitute x into the equation and
perform the indicated operations:
 1 1  1 1 e3t   3e3t   e3t 

x  
 3t    3t   3 3t   x
 4 1  4 1 2e   6e   2e 
Homogeneous Case; Vector Function Notation
As in Chapters 3 and 4, we first examine the general
homogeneous equation x' = P(t)x.
Also, the following notation for the vector functions
x(1), x(2),…, x(k),… will be used:
 x11 (t ) 
 x12 (t ) 
 x1n (t ) 






 x21 (t )  ( 2)
 x22 (t ) 
 x2 n (t ) 
(1)
(k )
x (t )  
, x (t )  
,, x (t )  
,












 x (t ) 
 x (t ) 
 x (t ) 
 n1 
 n2 
 nn 
Theorem 7.4.1
If the vector functions x(1) and x(2) are solutions of the system
x' = P(t)x, then the linear combination c1x(1) + c2x(2) is also a
solution for any constants c1 and c2.
Note: By repeatedly applying the result of this theorem, it
can be seen that every finite linear combination
x  c1x (1) (t )  c2 x ( 2) (t )    ck x ( k ) (t )
of solutions x(1), x(2),…, x(k) is itself a solution to x' = P(t)x.
Example 2
Consider the homogeneous equation x' = P(t)x below, with
the two solutions x(1) and x(2) as indicated.
3t
t




e
e
 1 1
(1)
( 2)

x ; x (t )   3t , x (t )  
x  
t 
 4 1
 2e 
  2e 
Then x = c1x(1) + c2x(2) is also a solution:
 1 1
 1 1 c1e 3t   1 1 c2 e t 



x  


3t  
t 
 4 1
 4 1 2c1e   4 1  2c2 e 
 3c1e 3t    c2 e t 


 
3t  
t 
 6c1e   2c2 e 
 3e 3t 
  e t 
 c1  3t   c2  t   x
 6e 
 2e 
Theorem 7.4.2
If x(1), x(2),…, x(n) are linearly independent solutions of the
system x' = P(t)x for each point in I:  < t < , then each
solution x = (t) can be expressed uniquely in the form
x  c1x (1) (t )  c2 x ( 2) (t )    cn x ( n ) (t )
If solutions x(1),…, x(n) are linearly independent for each
point in I:  < t < , then they are fundamental solutions
on I, and the general solution is given by
x  c1x (1) (t )  c2 x ( 2) (t )    cn x ( n ) (t )
The Wronskian and Linear Independence
The proof of Thm 7.4.2 uses the fact that if x(1), x(2),…, x(n)
are linearly independent on I, then detX(t)  0 on I, where
 x11 (t )  x1n (t ) 


X(t )   

 ,
 x (t )  x (t ) 
nn
 n1

The Wronskian of x(1),…, x(n) is defined as
W[x(1),…, x(n)](t) = detX(t).
It follows that W[x(1),…, x(n)](t)  0 on I iff x(1),…, x(n) are
linearly independent for each point in I.
Theorem 7.4.3
If x(1), x(2),…, x(n) are solutions of the system x' = P(t)x on
I:  < t < , then the Wronskian W[x(1),…, x(n)](t) is either
identically zero on I or else is never zero on I.
This result enables us to determine whether a given set of
solutions x(1), x(2),…, x(n) are fundamental solutions by
evaluating W[x(1),…, x(n)](t) at any point t in  < t < .