# 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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