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⎧
−utt + 3uxx = 0 ]0, π[×]0, +∞[
⎪⎨
ux(0, t) = ux(π, t) = 0
u(x, 0) = 0
⎪⎩
ut(x, 0) = x.
u(x, t) = T (t)X(x) ˙ X(0) = ˙ X(π) = 0
λ ∈ R
− ¨ T (t)X(x) + 3T (t) ¨ X(x) = 0,
T (t)X(x)
− ¨ T (t)
T (t) = −3 ¨ X(x)
= λ.
X(x)
− ¨ T (t) − λT (t) = 0,
3 ¨ X(x) + λX(x) = 0.
X(x) X(0) = X(π) = 0
3µ 2 + λ = 0 ∆ = −12λ ∆ > 0 λ < 0
X(x) = c1e µ1x +
c2e µ2x ˙ X(x) = c1µ1e µ1x + c2µ2e µ2x ˙ X(0) = 0 ˙ X(π) = 0
c1 c2
c1µ1 + c2µ2 = 0
c1µ1e µ1π + c2µ2e µ2π = 0
µ1µ2(e µ2π − e µ1π ) = 0 c1 = c2 = 0
∆ = 0 λ = 0 µ1 = 0
X(x) = c1 + c2x ˙ X(x) = c2
c2 = 0 X(x) = c1 c1 ∈ R \ {0}
∆ < 0 λ > 0 ±iω ω =
λ/3 X(x) = c1 cos(ωx)+c2 sin(ωx) ˙ X(x) = −ωc1 sin(ωx)+
ωc2 cos(ωx) ˙ X(0) = 0 c2 = 0 X(x) = c1 cos(ωx) ˙ X(π) = 0
ω ∈ N ∆ < 0 ω = 0 λ = 3n 2 n = 0
Xn(x) = cn cos(nx) n ∈ N ∆ ≤ 0
T λ = 3n 2 n ∈ N ¨ T (t) + 3n 2 T (t) = 0
T (0) = 0 n = 0 U(t) = c1 + c2t
U(t) = c2t n = 0 µ 2 + 3n 2 = 0
µ1 = in √ 3 µ2 = −in √ 3
T (t) = c1 cos(n √ 3t) + c2 sin(n √ 3t) T (0) = 0 c1 = 0 Tn(t) = dn sin(n √ 3t)
kn = cndn
u0(x, t) = U(t)X(x) = k0t
un(x, t) = U(t)X(x) = kn sin(n √ 3t) cos(nx)
∂tu0(x, 0) = k0
∂tun(x, 0) = n √ 3kn cos(nx)