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∆ = 0 λ = −1 µ1 = µ2 = −1
Φ(c1, c2, x) = c1e−x + c2xe−x
c1 = c2 = 0
√
|∆|
∆ < 0 α = −1 ω = 2 Φ(c1, c2, x) =
e−x (c1 cos ωx + c2 sin ωx)
c1 = 0 Φ(c1, c2, 0) =
X(0) = 0 Φ(c1, c2, π) = X(π) = 0 ω = n ∈ Z
ω ∆ ∆ < 0
−4n2 = 4(1 + λ) λ λn = −1 − n2
Xn(x) = cne −x sin(nx).
U λ
− ˙ T (t) + −3 − 1 − n 2 T (t) = 0.
Tn(t) = un(0)e −(4+n2 )t
X U λn
bn = cnun(0)
un(t, x) = bne −(4+n2 )t e −x sin(nx).
u(0, x) = xe −x ∞
= bne −x sin(nx) = e −x
∞
bn sin(nx),
n=1
bn x
x ∈ [0, π] [−π, π] 2π R
bn = 2
π
π
= 2
π
0
−π
x sin nx dx = 2
−
π
cos nπ
n
u(t, x) = e −x
x=π
x cos(nx)
n x=0
1
+ sin(nπ) =
n2 2(−1)n+1
n
∞
n=1
n=1
2(−1) n+1
e −(4+n2 )t
sin(nx).
n
+ 1
π
cos nx dx
n 0