Math 257/316 Assignment 3 Due Friday Jan 30 in class 1. The steady-state temperature distribution y(x) along a wire 0 ≤ x ≤ 1, cooled by its surroundings, and held fixed at zero temperature at its left endpoint, solves: d p(x)y 0 − y = 0, y(0) = 0, dx where p(x) ≥ 0 represents its (spatially dependent) thermal conductivity. If the conductivity degenerates at the left endpoint as p(x) = xs for some 0 ≤ s < 1, how does the temperature behave near that point? (I.e. just give the form of the the first term in a series solution about x0 = 0 (for x > 0).) Does the problem have a solution if s = 1? s > 1? 2. Find (the first three non-zero terms of) a series solution (about x0 = 0) of this initial value problem for a spherical Bessel equation: x2 d2 y dy 5 + 2x + (x2 − )y = 0, dx2 dx 16 y(0) = 0, lim x3/4 y 0 (x) = 1. x→0+ 3. In the notes, one solution of the Bessel equation of order zero x2 y 00 + xy 0 + x2 y = 0 is found to be J0 (x) = 1 − ∞ X 1 4 1 (−1)m 2m 1 2 6 x + x − x + · · · = x . 22 22 42 22 42 62 22m (m!)2 m=0 Find a second (linearly independent) solution of the form y2 (x) = J0 (x) ln(x) + ỹ(x) as follows: (a) show that for y2 to solve the Bessel equation, ỹ must solve the ode x2 ỹ 00 + xỹ 0 + x2 ỹ + 2xJ00 (x) = 0; (b) find the first three non-zero terms of a series solution ỹ(x) = ode. 1 P∞ n=1 bn x n of this