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1 Use the Taylor series method described below Theorem 3.3.1 to solve the initial- value problem y" + xy = 0 with y(0) = 2 and y'(0) = 1. Your solution should include all terms up to fifth-order. 2 Find the largest interval on which the problem y'''' + xy" + x2y + 3y' + xy = 0 with y (2) = y'(2) = y"(2) = 0 and y'"(2) = 1 is sure to have a unique solution. Explain why your answer is correct. 3 Find the general solution of the DE and the particular solution which satisfies the initial conditions. (a) y" - 2y + 2y = 0 with y(0) = 0 and y'(0) = 4 (b) y''' + 5y" = 0 with y(0) = 2 and y' (0) = y"(0) = 0.

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