We need to consider three cases as it turns out only

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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 d 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 d2 u = 0, 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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