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[0, π]
utt + 2ut − uxx = 0, ux(0, t) = ux(π, t) = 0,
u(x, 0) = 0 ut(x, 0) = x
u(x, t) = T (t)X(x)
¨T (t)X(x) + 2 ˙ T (t)X(x) − T (t) ¨ X(x) = 0
T (t)X(x)
¨T (t) + 2 ˙ T (t)
−
T (t)
¨ X(x)
= 0,
X(x)
¨X(x) − λX(x) = 0
¨T (t) + 2 ˙ T (t) − λT (t) = 0,
ux(0, t) = T (t) ˙ X(0) = 0 ux(π, t) = T (t) ˙ X(π) = 0 ˙ X(0) = ˙ X(π) = 0
¨X(x) − λX(x) = 0
˙X(0) = ˙ X(π) = 0,
λ ∈ R µ 2 = λ
λ > 0
√
λ x
X(x) = c1e + c2e −√λ x
, c1, c2 ∈ R
√ √ √
λ x −
λe − c2 λe √ λ x
, c1, c2 ∈ R
˙X(x) = c1
0 = ˙ X(0) = (c1 − c2) √ λ c1 = c2 0 = ˙ X(π) =
√ √
c1 λ(e λπ − e− √ λπ ) c1 = 0
λ = 0 X(x) = c1 + c2x c1, c2 ∈ R ˙ X(x) = c2 c2 = 0
X(x) = c1 ∈ R \ {0}
λ < 0 ω = |λ| X(x) = c1 cos(ωx) + c2 sin(ωx) c1, c2 ∈ R
˙ X(x) = −c1ω sin(ωx) + c2ω cos(ωx) 0 = ˙ X(0) = c2ω c2 = 0
0 = ˙ X(π) = −c1ω sin(ωx) ω ∈ Z λ = −n 2 n ∈ N n = 0
X(x) λ = −n 2 n ∈ N Xn(x) =
cn cos(nx) λ = n = 0 T (t)
¨T (t) + 2 ˙
T (t) + n 2 T (t) = 0,
T (0) = 0.