Lecture Notes in Differential Equations - Bruce E. Shapiro

323
where (A − λI)w 2 = w 1 , i.e.,
Letting w 2 =
{( 3 3
0 3
) ( 1 0
− 3
0 1
(
c
d)
, this simplifies to
)} ( 1
w 2 =
0
)
(31.159)
( 0 3
0 0
) ( c
d
) ( 1
=
0
)
(31.160)
which gives d = 1/3 and leaves c ( undetermined ) (namely, any value will do,
0
so we choose zero). Hence w 2 = :
1/3
(
y 2 = e
[t
3t 1
0
The general solution is then
) ( 0
+
1/3
)]
= e 3t ( t
1/3
)
(31.161)
y = c 1 y 1 + c 2 y 2 (31.162)
( ) ( )
1
= c 1 e 3t t
+ c
0
2 e 3t (31.163)
1/3
where c 1 and c 2 are arbitrary constants. qed
Variation of Parameters for Systems
Theorem 31.14. A particular solution of
y ′ = Ay + g(t) (31.164)
is
y P = e At ∫
e −At g(t)dt (31.165)
Proof. Let W be a fundamental matrix of the homogeneous equation
y ′ = Ay, and denote a fundamental set of solutions by y 1 , ..., y n . We will
look for a particular solution of the form
y P =
n∑
y i (t)u i (t) = Wu (31.166)
i=1