3 for us however the main point of the method of nite

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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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