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Mathematics 238 test three due Friday, May 27, 2005 1. Given the differential equation (x2 + 4)y 00 + 3xy 0 + 4y = 0 • list the singular points of the equation. ∞ • for which x-values will a power series solution of the form X a (x−3)k converge? k k=0 2. For which values of x does the each of following power series converge? (a) ∞ X (x − 4)2k k=1 (b) ∞ X k9k a2k x2k where a2 = 1 & ka2k+2 = 3(k + 1)a2k for k ≥ 1 k=1 3. Find a power series solution about x = 0 (through the degree four term) to the first-order initial value problem dy = x2 + y 2 dx and y(0) = 1 extra credit parts of problem (3) • compare your solution to a numerically obtained solution from a math software package or website. (plot the two solutions together and compare) • compare your solution to the first few Picard iterates. 4. Given the second-order differential equation y 00 + xy 0 + y = 0 • for which values of x will power series solutions of this differential equation converge? • find the recursion relation for the power series solution about x = 0. • write the general solution through the degree five terms. • find the solution (through degree five terms) to the initial value problem y(0) = 0 and y 0 (0) = 1. • estimate the accuracy of your solution of the previous part when −0.5 ≤ x ≤ 0.5. Mathematics 238 test three, part two due Monday, June 6, 2005 5. Use the elimination method to solve the system of equations d2 x1 dt2 d2 x2 dt2 = x2 − x1 − 1 = −(x2 − x1 − 1) + (x3 − x2 − 1) d2 x 3 dt2 = −(x3 − x2 − 1) Begin the process by rewriting the system in the format (D2 + 1)x1 − x2 = −1 −x1 + (D2 + 2)x2 − x3 = 0 −x2 + (D2 + 1)x3 = 1