MATH 257 Study Guide - Midterm Guide: Briey, Boundary Value Problem, Heat Equation
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Page 2 of 17 pages: consider the differential equation (a) classify the points 0 x < as ordinary points, regular singular points, or irregular singular points. 8x2y00 + 2xy0 + (1 + 2x)y = 0 (1) (b) Find two values of r such that there are solutions of the form y(x) = anxn+r. Pn=0 (c) use the series expansion in (b) to determine two independent solutions of (1). You only need to calculate the rst three non-zero terms in each case. Page 3 of 17 pages (question 1 continued) Page 4 of 17 pages (question 1 continued) Page 5 of 17 pages: consider the following initial boundary value problem for the heat equation: ut = uxx u , 0 < x < 1 , t > 0 ux(0, t) = 0, u(x, 0) = x ux(1, t) = 0 (a) determine the solution to the boundary value problem (2) by separation of variables. (2)
