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Unformatted text preview: log x),
where log denotes the natural or base e logarithm. The lemma implies that
these eigenfunctions will be perpendicular with respect to the inner product
·, · , deﬁned by
eπ f, g =
1 1
f (x)g (x)dx.
x Exercises:
4.7.1. Show by direct integration that if m = n, the functions
fm (x) = sin(m log x) and fn (x) = sin(n log x) are perpendicular with respect to the inner product deﬁned by (4.33). 110 (4.33) 4.7.2. Find the function u(x, t), deﬁned for 1 ≤ x ≤ eπ and t ≥ 0, which satisﬁes
the following initialvalue problem for a heat equation with variable coeﬃcients:
∂u
∂
=x
∂t
∂x x ∂u
∂x u(1, t) = u(eπ , t) = 0, , u(x, 0) = 3 sin(log x) + 7 sin(2 log x) − 2 sin(3 log x).
4.7.3.a. Find the solution to the eigenvalue problem for the operator
L=x d
dx x d
dx − 3, which acts on the space V0 of functions f : [1, eπ ] → R which vanish at the
endpoints of the interval [1, eπ ].
b. Find the function u(x, t), deﬁned for 1 ≤ x ≤ eπ and t ≥ 0, which satisﬁes
the following initialvalue problem for a heat equation...
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 Winter '14
 Equations

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