15.3.2 System of Differential Equations in a Matrix Form:
Resolvent Matrix
The equations of (15.54) and (15.55) are unified in a homogeneous single matrix
equation such that
ð _
x t
ð Þ _
y t
ð ÞÞ ¼ x t
ð Þ y t
ð Þ
ð
Þ
a t
ð Þ c t
ð Þ
b t
ð Þ d t
ð Þ
:
ð15:66Þ
In (15.66), x(t) and y(t) and their derivatives represent the functions (of t), and so
we describe them as row matrices. With this expression of equation, see Sect. 11.2
and, e.g., (11.37) therein. Another expression is a transposition of (15.66) described
by
_
x t
ð Þ
_
y t
ð Þ
!
¼
a t
ð Þ b t
ð Þ
c t
ð Þ d t
ð Þ
x t
ð Þ
y t
ð Þ
:
ð15:67Þ
Notice that the coefficient matrix has been transposed in (15.67) accordingly. We
are most interested in an equation of constant coefficient and, hence, we rewrite
(15.66) once again explicitly as
ð _
x t
ð Þ _
y t
ð ÞÞ ¼ x t
ð Þ y t
ð Þ
ð
Þ
a c
b d
ð15:68Þ
or
d
dt
x t
ð Þ y t
ð Þ
ð
Þ¼ x t
ð Þ y t
ð Þ
ð
Þ
a c
b d
:
ð15:69Þ
Putting F(t) (x(t) y(t)), we have
dF t
ð Þ
dt
¼ F t
ð Þ
a c
b d
:
ð15:70Þ
This is exactly the same as (15.50) if we put
A ¼
a c
b d
:
ð15:71Þ
Then, we get
F t
ð Þ ¼ exp tA:
ð15:72Þ
15.3 System of Differential Equations
595