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Unformatted text preview: tudy of heat ﬂow often leads to “boundary-value problems” for ordinary diﬀerential equations. Indeed, in the “steady-state” case, in which u is
independent of time, equation (4.5) becomes
dx κ(x) du
(x) + µ(x)u(x) + ν (x) = 0,
dx 2 A description of the Black-Scholes technique for pricing puts and calls is given in Paul
Wilmott, Sam Howison and Jeﬀ Dewynne, The mathematics of ﬁnancial derivatives , Cambridge Univ. Press, 1995. 83 a linear ordinary diﬀerential equation with variable coeﬃcients. Suppose now
that the temperature is speciﬁed at the two endpoints of the bar, say
u(0) = α, u(L) = β. Our physical intuition suggests that the steady-state heat equation should have
a unique solution with these boundary conditions.
a. Solve the following special case of this boundary-value problem: Find u(x),
deﬁned for 0 ≤ x ≤ 1 such that
dx2 u(0) = 70, u(1) = 50. b. Solve the following special case of this boundary-value problem: Find u(x),
deﬁned for 0 ≤ x ≤...
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- Winter '14