9.17.2. Theorems 9.14-16.
Theorems 9.14-15 will show that (9.46a) always possesses a nonsingular solution.
Theorem 9.14.
Let A  t  be an n  n matrix function continuous on an open interval J.
Let a  J
and B be a constant n  m matrix. Then the homogeneous linear system
Y  t   At Y t 
with
Y a  B
has an n  m matrix solution Y on J.
Proof
See Theorem 10.3 of chapter 10 .
Theorem 9.15.
Let P  t  be an n  n matrix function continuous on an open interval J. Then there
exists an n  n matrix function F on J that satisfies
F   x   F  x  P  x 
with F  a   I
(9.47)
where a  J . Moreover, F  x  is nonsingular  x  J .
Proof
Let Yk  x  be an n  1 columnn matrix solution of the equation
Yk  x    P  x  Yk  x 
t
on J with initial condition Yk  a   I k , where Ik is the kth column of the n  n identity
matrix I.
By Theorem 9.14, Yk  x  always exists.
Let G be the n  n matrix
whose kth column is Yk. The G satisfies
G  x    P  x  G  x 
t
on J with initial condition G  a   I .
(9.48)
Taking the transpose of (9.48), we have
Gt  x   Gt  x  P  x 
Thus, the n  n matrix F  G t satisfies the initial problem (9.47).
Next, we prove that F is nonsingular by exhibiting its inverse.
matrix solution of the initial problem on J
H  x  P  x H  x
with
Let H be the n  n
H  a  I
Thus,
 F  x  H  x    F  x  H   x   F   x  H  x 
 F  x P  x H  x  F  x P  x H  x  O
 xJ
Hence, F  x  H  x  is constant so that
F  x  H  x   F a  H a   I
i.e., H is the inverse of F.
QED.
Theorem 9.16.
Given an n  n matrix function P and an n  m matrix function Q, both continuous on
an open interval J. The solution of the initial problem on J
Y  t   At Y t 
with
Y a  B
(9.49)
is given by the formula
Y  x  F
1
x
 x  B  F  x   dt F t  Q t 
1
(9.50)
a
where F is an n  n matrix given by F  x   Gt  x  , where G  x  satisfies
G  x    P  x  G  x 
t
with
G a  I
Proof
This is simply a summary of the results of §9.17.
(9.51)