Lecture Notes in Differential Equations - Bruce E. Shapiro

322 LESSON 31. LINEAR SYSTEMS
Theorem 31.13. Let λ 1 , λ 2 , ..., λ j be the eigenvalues of A corresponding
to distinct, linearly independent, eigenvectors v 1 , ..., v j . Denote the
multiplicity of eigenvalue λ i by k i , for i = 1, 2, ..., j (i.e., ∑ k i = n). Let
w i1 , · · · , w ij be the generalized eigenvectors for λ i , with w i1 = v i . Then a
fundamental set of solutions to y ′ = Ay are
Example 31.4. Solve
y i1 = w i1 e λit (31.149)
y i2 = (tw i1 + w i2 )e λit (31.150)
( )
t
2
y i3 =
2 w i1 + tw i2 + w i3 e λit (31.151)
.
y ij =
∑k i
m=1
y ′ =
t ki−m
(k i − m)! w me λit (31.152)
( 3 3
0 3
)
y (31.153)
The characteristic polynomial is
0 =
∣ 3 − λ 3
0 3 − λ ∣ = (3 − λ)2 (31.154)
( a
which has λ = 3 as a solution of multiplicity 2. Letting denote the
b)
eigenvector, ( 3 3
0 3
) ( a
b
)
( a
= 3
b
which can be decomposed into the following pair of equations
3a + 3b = 3a
3b = 3b
)
(31.155)
(31.156)
Hence there is only one eigenvector corresponding to λ = 3, namely (any
multiple of) (1 0) T . Therefore one solution of the differential equation is
( )
y 1 = ve λt 1
= e 3t (31.157)
0
and a second, linearly independent solution, is
y 2 = (tv + w 2 )e 3t (31.158)