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# m238sp05t3 - Mathematics 238 test three due Friday 1 Given...

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Mathematics 238 test three due Friday, May 27, 2005 1 . Given the differential equation ( x 2 + 4) y + 3 xy + 4 y = 0 list the singular points of the equation. for which x -values will a power series solution of the form k =0 a k ( x - 3) k converge? 2 . For which values of x does the each of following power series converge? ( a ) k =1 ( x - 4) 2 k k 9 k ( b ) k =1 a 2 k x 2 k where a 2 = 1 & ka 2 k +2 = 3( k + 1) a 2 k for k 1 3 . Find a power series solution about x = 0 (through the degree four term) to the first-order initial value problem dy dx = x 2 + y 2 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 + xy + 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

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