Unformatted text preview: x ≤ π and t ≥ 0, which satisﬁes
the following conditions:
∂u
∂2u
=
,
∂t
∂x2 u(0, t) = u(π, t) = 0, u(x, 0) = sin x + 3 sin 2x − 5 sin 3x. 4.2.3. Find the function u(x, t), deﬁned for 0 ≤ x ≤ π and t ≥ 0, which satisﬁes
the following conditions:
∂u
∂2u
, u(0, t) = u(π, t) = 0, u(x, 0) = x(π − x).
=
∂t
∂x2
(In this problem you need to ﬁnd the Fourier sine series of h(x) = x(π − x).)
4.2.4. Find the function u(x, t), deﬁned for 0 ≤ x ≤ π and t ≥ 0, which satisﬁes
the following conditions:
1 ∂u
∂2u
,
=
2t + 1 ∂t
∂x2 u(0, t) = u(π, t) = 0, u(x, 0) = sin x + 3 sin 2x. 4.2.5. Find the function u(x, t), deﬁned for 0 ≤ x ≤ π and t ≥ 0, which satisﬁes
the following conditions:
(t + 1) ∂u
∂2u
,
=
∂t
∂x2 u(0, t) = u(π, t) = 0, u(x, 0) = sin x + 3 sin 2x. 4.2.6.a. Find the function w(x), deﬁned for 0 ≤ x ≤ π , such that
d2 w
= 0,
dx2 w(0) = 10, w(π ) = 50. b. Find the general solution to the following boundary value problem for the
heat equation: Find the functions u(x...
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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