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Unformatted text preview: Problem 1 (17pts) . Solve y 00 + xy + 2 y = 0 , y (0) = 1 , y (0) = 0 , by means of a power series about x = 0. Find the recurrence relation and compute the first three nonzero terms. Find the closed formula for the coefficients and compute the radius of convergence of the power series. 1 Problem 2 (10pts) . Consider the following 2nd order differential equation (2 x 2 x + 1) y 00 + ln(1 + x ) y + cos ( sin( x ) ) y = xe x 2 y (0) = y (0) = 1 . (1) Is the solution of equation (1) analytic around x = 0? If so, what is the optimal lower bound for the radius of convergence of the power series of y at 0 you can guarantee? Also, find the polynomial of degree 3 that best approximates y in a neighborhood of 0. 2 Problem 3 (6pts + 8pts + 8pts) . Consider the Euler equation x 2 y 00 + αxy + βy = 0 , x > . (2) and let F ( r ) = r ( r 1) + αr + β . From item (a) to item (c), justify every single step of your solution. Show all the work....
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 Spring '10
 Field
 Calculus, Derivative, Power Series, Laplace, linearly independent solutions

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