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Math 257 – Assignment 2 Due: Wednesday, January 19 1. For the following differential equations, verify that x0 = 0 is an ordinary point. Without actually solving the equations, find then a lower bound for the radii of convergence of power series solutions about x0 = 0: a) (x2 + 2)y 00 + 2xy 0 = 0 b) (x − 1)y 00 + y 0 = 0 c) xy 00 + sin(x)y = 0 Hint: sin(x) = x − x3 x5 x7 + − ± ... 3! 5! 7! 2. For the following differential equations, verify that 0 is an ordinary point. Then find two linearly independent series solution for each problem and determine their guaranteed radius of convergence: a) (x2 − 3)y 00 + 2xy 0 = 0 b) y 00 − xy 0 − x2 y = 0 3. Consider the equation x2 y 00 + x2 y 0 + y = 0. Show that: a) x0 = 0 is a singular point. b) The equation has no non-trivial power series solution of the form y = Hint: Show that an = 0 for all n ≥ 0. P∞ n=0 an x n . 4. Find the first four non-zero terms to the series solution of the equation (x2 − 4)y 00 + 3xy 0 + y = 0, y(0) = 5, y 0 (0) = 1. 5. Find a power series solution of the non-homogeneous equation y 00 − xy = 1 about the ordinary point x0 = 0.