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6.3 Homogeneous Systems
In this section we give the basic theory for linear homogeneous systems. This “theory” is simply a
repetition results given in Sections 3.2 and 6.1, phrased this time in terms of the system
x
′ 1 =^ a^11 (t)x^1 +^ a^12 (t)x^2 +^ · · ·^ +^ a^1 n(t)xn(t)
x
′ 2 =^ a^21 (t)x^1 +^ a^22 (t)x^2 +^ · · ·^ +^ a^2 n(t)xn(t)
. . .
x ′ n = an 1 (t)x 1 + an 2 (t)x 2 + · · · + ann(t)xn(t)
(H)
or
x
′ = A(t)x. (H)
Note first that the zero vector z(t) ≡ 0 =
is a solution of (H). As before, this solution
is called the trivial solution. Of course, we are interested in finding nontrivial solutions.
THEOREM 1. If v is a solution of (H) and α is any real number, then u = αv is also a
solution of (H); any constant multiple of a solution of (H) is a solution of (H).
THEOREM 2. If v 1 and v 2 are solutions of (H), then u = v 1 + v 2 is also a solutions of (H);
the sum of any two solutions of (H) is a solution of (H).
These two theorems can be combined and extended to:
THEOREM 3. If v 1 , v 2 ,... , vk are solutions of (H), and if c 1 , c 2 ,... , ck are real numbers,
then
c 1 v 1 + c 2 v 2 + · · · + ckvk
is a solution of (H); any linear combination of solutions of (H) is also a solution of (H).
DEFINITION 1. Let
v 1 =
v 11 (t)
v 21 (t)
. . .
vn 1 (t)
, v 2 =
v 12 (t)
v 22 (t)
. . .
vn 2 (t)
,... , vk =
v 1 k(t)
v 2 k(t)
. . .
vnk(t)
be n-component vector functions defined on some interval I. The vectors are linearly dependent on
I if there exist k real numbers c 1 , c 2 ,... , ck, not all zero, such that
c 1 v 1 (t) + c 2 v 2 (t) + · · · + ck vk(t) ≡ 0 on I.
Otherwise the vectors are linearly independent on I.
THEOREM 4. Let v 1 , v 2 ,... , vn be n, n-component vector functions defined on an interval
I. If the vectors are linearly dependent, then
v 11 v 12 · · · v 1 n
v 21 v 22 · · · v 2 n
. . .
vn 1 vn 2 · · · vnn
≡ 0 on I.
The determinant in Theorem 4 is called the Wronskian of the vector functions v 1 , v 2 ,... , vn.
Example 1. The vector functions
u =
t 3
3 t
2
and v =
t − 1
−t
− 2
are solutions of the homogeneous system in Example 3, Section 6.2. Their Wronskian is:
W (x) =
t
3 t
− 1
3 t
2 −t
− 2
= − 4 t.
The vector functions
v 1 =
e
2 t
2 e 2 t
4 e
2 t
,^ v 2 =
e
− 3 t
− 3 e − 3 t
9 e
− 3 t
,^ v 3 =
te
2 t
e 2 t
4 e
2 t
2 t
are solutions of the homogeneous system
x
′
x.
Their Wronskian is:
W (x) =
e
2 t e
− 3 t te
2 t
2 e
2 t − 3 e− 3 t e
2 t
2 t
4 e 2 t 9 e − 3 t 4 e 2 t
= − 25 e
t .
THEOREM 5. Let v 1 , v 2 ,... , vn be n solutions of (H). Exactly one of the following holds:
- W (v 1 , v 2 ,... , vn)(t) ≡ 0 on I and the solutions are linearly dependent.
- W (v 1 , v 2 ,... , vn)(t) 6 = 0 for all t ∈ I and the solutions are linearly independent.
It is easy to construct sets of n linearly independent solutions of (H). Simply pick any point
a ∈ I and any nonsingular n × n matrix A. Let α 1 be the first column of A, α 2 the second
column of A, and so on. Then let v 1 be the solution of (H) such that v 1 (a) = α 1 , let v 2 be the
solution of (H) such that v 2 (a) = α 2 ,.. ., and let vn be the solution of (H) such that vn = αn.
The existence and uniqueness theorem guarantees the existence of these solutions. Now
W (v 1 , v 2 ,... , vn)(a) = det A 6 = 0.
- u =
cos t
sin t
, v =
sin t
cos t
- u =
t − t
2
−t
, v =
− 2 t + 4t
2
2 t
- u =
te t
t
, v =
e t
- u =
2 − t
t
,^ v^ =
t
,^ w^ =
2 + t
t − 2
- u =
cos t
sin t
, v =
cos t
sin t
, w =
cos t
sin t
- u =
e
t
−e
t
e
t
,^ v^ =
−e
t
2 e
t
−e
t
,^ w^ =
e
t
- u =
2 − t
t
, v =
t + 1
, w =
t
t + 2
- u =
e t
, v =
, w =
e t
- u =
cos (t + π/4)
, v =
cos t)
e t
, w =
sin t)
e t
- Given the linear differential system
x
′
x.
Let
u =
e
2 t
e
2 t
and v =
3 e
3 t
2 e
3 t
(a) Show that u, v are a fundamental set of solutions of the system.
(b) Let V be the corresponding fundamental matrix. Show that
V
′ = AV.
(c) Give the general solution of the system.
(d) Find the solution of the system that satisfies x(0) =
- Let V be the matrix function
V (t) =
cos 2t sin 2t
sin 2t − cos 2t
(a) Verify that V is a fundamental matrix for the system
x
′
x.
(b) Find the solution of the system that satisfies x(0) =
- Let V be the matrix function
V (t) =
0 4 te
−t e
−t
1 e
−t 0
(a) Verify that V is a fundamental matrix for the system
x
′
x.
(b) Find the solution of the system that satisfies x(0) =
