Let ux t vertical displacement of the string at the

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Unformatted text preview: , t), defined for 0 ≤ x ≤ π and t ≥ 0, such that ∂u ∂2u , = ∂t ∂x2 u(0, t) = 10, 90 u(π, t) = 50. (4.13) (Hint: Let v = u − w, where w is the solution to part a. Determine what conditions v must satisfy.) c. Find the particular solution to (4.13) which in addition satisfies the initial condition 40 u(x, 0) = 10 + x + 2 sin x − 5 sin 2x. π 4.2.7.a. Find the eigenvalues and corresponding eigenfunctions for the differential operator d2 L = 2 + 3, dt which acts on the space V0 of well-behaved functions f : [0, π ] → R which vanish at the endpoints 0 and π by L(f ) = d2 f + 3f. dt2 b. Find the function u(x, t), defined for 0 ≤ x ≤ π and t ≥ 0, which satisfies the following conditions: ∂u ∂2u + 3u, = ∂t ∂x2 u(0, t) = u(π, t) = 0, u(x, 0) = sin x + 3 sin 2x. 4.2.8. The method described in this section can also be used to solve an initial value problem for the heat equation in which the Dirichlet boundary condition u(0, t) = u(L, t) = 0 is replaced by the...
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