Example 10-11. Represent the given system of differential equations in matrix
form.
dy 1
dt=y 1 +y 2 −y 3 + sin t, dy^2
dt= 2y 2 +y 3 + cos t, dy^3
dt= 3y 3 + sin2tSolution The above system of differential equations can be represented in the form
dy
dt=Ay+f(t) (10.8)where y =y(t) = col(y 1 , y 2 , y 3 )denotes a column vector, A=
1 1 − 1
0 2 1
0 0 3is a co-
efficient matrix and f =f(t) = col(sin t,cos t, sin 2 t) represents a variable right-hand
side to the differential system. Matrix differential equations of the form given by
equation (10.8) subject to the initial condition y(0) = c, where cis a constant, are
called initial-value problems.
Example 10-12. The nth order linear differential equation
dny
dtn+a 1 (t)dn− (^1) y
dtn−^1
+a 2 (t)d
n− (^2) y
dtn−^2
+···+an− 2 (t)d
(^2) y
dt^2
+an− 1 (t)dy
dt
+an(t)y= 0
is converted to matrix form by defining
y ̄=col(y, dydt ,d
(^2) y
dt^2 ,.. .,
dn−^2 y
dtn−^2 ,
dn−^1 y
dtn−^1 )
and
A=A(t) =
0 1 0 0 ··· 0 0
0 0 1 0 ··· 0 0
0 0 0 1 ··· 0 0..
.
..
.
..
.
..
. ...
..
.
..
.
0 0 0 0 ··· 1 0
0 0 0 0 ··· 0 1
−an(t) −an− 1 (t) −an− 2 −an− 3 (t) ···− a 2 (t) −a 1 (t)
The given scalar equation can then be represented by the matrix equation
d ̄y
dt =A(t) ̄y