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 vector-valued 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 complex-valued differential equation can be obtained by taking the real and imaginary parts and solving a system of the form Equation 1.
