# a2 - radius of convergence a x 2-3 y 00 2 xy = 0 b y...

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Math 257/316 – Assignment 2 Due: Wednesday, January 20 1. The following functions are analytic at x 0 = 0 . Without actually computing their deriva- tives, ﬁnd the ﬁrst three non-zero terms of their power series about x 0 and determine their convergence radii: a) f ( x ) = 1 2 + x b) 1 cos x Hint: cos x = 1 - x 2 2! + x 4 4! - x 6 6! ± . . . 2. For the following equations, verify that x 0 = 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 x 0 = 0 : a) ( x 2 + 2) y 00 + 2 xy 0 = 0 b) ( x - 1) y 00 + y 0 = 0 c) xy 00 + sin( x ) y = 0 Hint: sin( x ) = x - x 3 3! + x 5 5! - x 7 7! ± . . . 3. For the following differential equations, verify that x 0 = 0 is an ordinary point. Then ﬁnd two linearly independent series solution for each problem and determine their guaranteed
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Unformatted text preview: radius of convergence: a) ( x 2-3) y 00 + 2 xy = 0 b) y 00-xy-x 2 y = 0 4. Consider the equation x 2 y 00 + x 2 y + y = 0 . Show that: a) x = 0 is a singular point. b) The equation has no non-trivial power series solution of the form y = ∑ ∞ n =0 a n x n . Hint: Show that a n = 0 for all n ≥ . 5. Find the ﬁrst four non-zero terms to the series solution of the equation ( x 2-4) y 00 + 3 xy + y = 0 , y (0) = 5 , y (0) = 1 . 6. Find a power series solution of the non-homogeneous equation y 00-xy = 1 about the ordinary point x = 0 ....
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