Note that the definition in Equation 1 encompasses nthnth order differential equations (i.e., equations that contains derivatives up to order nn), as well as complex valued and vectorvalued differential equations. Indeed, we can rewrite any nthnth order differential equation of the form:
u
(
n
)
(
t
)
=
F
(
t
,
u
(
t
)
,
u
'
(
t
)
,
...
,
u
(
n

1
)
(
t
)
)
u
(
n
)
(
t
)
=
F
(
t
,
u
(
t
)
,
u
'
(
t
)
,
...
,
u
(
n

1
)
(
t
)
)
(2)where u(i)=diu/dtiu(i)=diu/dti, in the form Equation 1 by letting:
x
≜
(
x
1
,
x
2
,
...
,
x
n
)
=
(
u
,
u
(
1
)
,
...
,
u
(
n

1
)
)
f
=
(
x
2
,
...
,
x
n
,
F
)
.
x
≜
(
x
1
,
x
2
,
...
,
x
n
)
=
(
u
,
u
(
1
)
,
...
,
u
(
n

1
)
)
f
=
(
x
2
,
...
,
x
n
,
F
)
.
(3)
Then, we have:
x
˙
(
t
)
=
d
d
t
u
(
t
)
u
(
1
)
(
t
)
⋮
u
(
n

1
)
(
t
)
=
u
(
1
)
(
t
)
u
(
2
)
(
t
)
⋮
u
(
n
)
(
t
)
=
x
2
(
t
)
⋮
x
n
(
t
)
F
(
t
,
x
(
t
)
)
=
f
(
t
,
x
(
t
)
)
.
x
˙
(
t
)
=
d
d
t
u
(
t
)
u
(
1
)
(
t
)
⋮
u
(
n

1
)
(
t
)
=
u
(
1
)
(
t
)
u
(
2
)
(
t
)
⋮
u
(
n
)
(
t
)
=
x
2
(
t
)
⋮
x
n
(
t
)
F
(
t
,
x
(
t
)
)
=
f
(
t
,
x
(
t
)
)
.
(4)Finally, note that the solution to a complexvalued differential equation can be obtained by taking the real and imaginary parts and solving a system of the form Equation 1.