Graphics Reference
In-Depth Information
The equation in Problem 2 is called the
nth-order differential equation
associated
to the function f and the n times differentiable function j(t), if it exists, is called a
solution
to the equation.
D.3.2. Theorem.
Let (t
0
,x
1
,x
2
,...,x
n
) Œ
D
. Then there exists an e>0 and unique n
times continuously differentiable function j :(t
0
-e,t
0
+e) Æ
R
that is the solution to
Problem 2 and satisfies j
(i)
(t
0
) = x
i+1
, 0 £ i < n.
Proof.
To prove the theorem one reduces Problem 2 to Problem 1 by introducing
new function j
i
and solving
¢
()
=
()
j
t
j
t
1
2
¢
()
=
()
j
t
j
t
2
3
M
M
()
=
()
j
¢
t t
tft t
j
n
-
1
n
¢
()
=
(
()
()
()
)
j
,
j
,
j
t
,...,
j
t
.
n
1
2
n
See [CodL55].
The values t
0
, x
1
, x
2
,..., and x
n
in Theorem D.3.1 and D.3.2 are called
initial con-
ditions
. The theorems can be rephrased as saying that initial conditions specify a
unique
local
solution. An interesting question is whether these local solutions extend
to global solutions. In two important special cases this is indeed the case.
Consider the
linear system of differential equations of the first order
¢
()
=
()
+
()
()
yx a xy a xy
+◊◊◊+
a xy
,
1
11
1
12
2
1
n
n
¢
()
=
()
+
()
()
yx a xy a xy
+◊◊◊+
a xy
,
2
21
1
22
2
2
n
n
M
M
M
M
¢
()
=
()
+
()
()
(D.7)
yx a xy a xy
+◊◊◊+
a xy
.
n
n
1
1
n
2
2
nn
n
Assume that the functions a
ij
(x) are continuous over some interval
X
which could be
open, closed, or all of
R
.
D.3.3. Theorem.
Let x
0
Œ
X
and let c
0
, c
1
,..., c
n-1
be arbitrary real numbers. There
exist unique functions y
i
(x) defined on
X
with continuous derivatives satisfying Equa-
tions (D.7) and the conditions
()
=
()
=
()
=
yx
c
,
y x
c
, ... ,
y x
c
.
10
1
20
2
n
0
n
Proof.
See [CodL55].
Definition.
A
linear differential equation
is any equation of the form
()
n
(
n
-
1
)
()
()
+
()
()
+ ◊◊◊+
()
¢
()
+
()()
=
()
a xy
x
a xy
x
a
xy x
a xyx
fx
,
(D.8)
0
1
n
-
1
n
where a
i
(x), y(x), f(x) are functions and y
(i)
(x) denotes the ith derivative of y(x).
We shall assume that the functions a
i
(x) and f(x) in Equation (D.8) are continuous
over some interval
X
which could be open, closed, or all of
R
and that a
0
(x) π 0 for x Œ
X
.
