# Just as before we substitute ux t f xg t into 428

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Unformatted text preview: λ = −n2 , f (x) = bn sin nx, where the bn ’s are constants. 101 for n = 1, 2, 3, . . . , 4.5.3. Find the function u(x, t), deﬁned for 0 ≤ x ≤ π and t ≥ 0, which satisﬁes the following conditions: ∂2u ∂2u = , ∂t2 ∂x2 u(0, t) = u(π, t) = 0, u(x, 0) = sin x + 3 sin 2x − 5 sin 3x, ∂u (x, 0) = 0. ∂t 4.5.4. Find the function u(x, t), deﬁned for 0 ≤ x ≤ π and t ≥ 0, which satisﬁes the following conditions: ∂2u ∂2u = , ∂t2 ∂x2 u(0, t) = u(π, t) = 0, ∂u (x, 0) = 0. ∂t u(x, 0) = x(π − x), 4.5.5. Find the function u(x, t), deﬁned for 0 ≤ x ≤ π and t ≥ 0, which satisﬁes the following conditions: ∂2u ∂2u = , 2 ∂t ∂x2 u(0, t) = u(π, t) = 0, u(x, 0) = 0, ∂u (x, 0) = sin x + sin 2x. ∂t 4.5.6. (For students with access to Mathematica) a. Find the ﬁrst ten coeﬃcients of the Fourier sine series for h(x) = x − x4 by running the following Mathematica program f[n ] := 2 NIntegrate[(x - x∧4) Sin[n Pi x], {x,0,1}]; b = Table[f[n], {n,1,10}] b. Find the ﬁrst ten terms of the solution to the initial value problem for a vi...
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## This document was uploaded on 01/12/2014.

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