pdf (it, 2.898,97 KB, 25/01/13)

uj n
y(t) = e (t−t0)A y0 +
t
t0
e (t−τ)A b(τ) dτ
y(0) = y0
n n ≥ 1
y (n) + an−1(t)y (n−1) + ... + a1(t)y ′ + a0(t)y = b(t)
b, a0, ..., an−1 ∈ C 0 (I, K) I R
y (n) + an−1(t)y (n−1) + ... + a1(t)y ′ + a0(t)y = 0
z1 = y z2 = y ′ zn = y (n−1) z ′ = A(t)z + B(t)
z ′ = A(t)z A(t) ∈ Matn×n(K) A
−a0(t), ..., −an−1(t)
1 0 B(t) b(t)
r ϕ1, ..., ϕr : I → K C m−1 I
m m × r K
i 0 m−1 ϕr r
n r
t0 r t ∈ I
n n ϕ1, ..., ϕn n
w(t)
n 0
wj(t) n n
j
n ϕ1, ..., ϕn
y (n) + an−1(t)y (n−1) + ... + a1(t)y ′ + a0(t)y = 0
y (n) + an−1(t)y (n−1) + ... + a1(t)y ′ + a0(t)y = b(t)
t0 ∈ I n − 1
n
ϕ0(t) = γj(t)ϕj(t)
w(t) wj(t)
γj(t) =
t
t0
j=1
(−1)
n+j wj(τ)
w(τ)
b(τ) dτ
n
y (n) + an−1y (n−1) + ... + a1y ′ + a0y = b(t),
y (n) + an−1y (n−1) + ... + a1y ′ + a0y = 0
p(z) = z n + an−1z n−1 + ... + a1z + a0 λ p(z) ν
e λt , te λt , ..., t ν−1 e λt
λ = αj + iβj ¯ λ = αj − iβj t k e λjt t k e ¯ λjt t k e αjt cos(βjt)
t k e αjt sin(βjt)
n = 2