a3 - Math 257/316 – Assignment 3 Due: Wednesday, January...

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Unformatted text preview: Math 257/316 – Assignment 3 Due: Wednesday, January 27 1. Find the first three non-zero terms in each of two linearly independent power series solutions centered at x = 0 of the equation: cos(x)y + xy − 2y = 0. What do you expect the convergence radius of the solutions to be? 2. For a constant λ, consider Hermite’s equation: y − 2xy + λy = 0, −∞ < x < ∞. a) Find the first four nonzero terms in each of two linearly independent series solutions centered at x0 = 0. b) If λ = 2n is a non-negative even integer, then one or the other of the series solution terminates and becomes a polynomial. Find these polynomials for λ = 8 and λ = 10. c) The Hermite polynomial Hn (x) is defined as the polynomial solution with λ = 2n for which the coefficient of xn is 2n . Find the polynomials H4 (x) and H5 (x). 3. For each of the following equations find all singular points and determine whether each one is regular or irregular: a) x3 y + 4(sin x)2 y = 0 c) xy + y + (cot x)y = 0 b) d) (x2 − 2x + 1)y + y = 0 x2 y + e1/x y = 0 2 4. Determine the general solutions of the following Euler equations that are valid in any interval not including the singular point: a) x2 y + 8xy + 12y = 0 b) (x − 2)2 y + 5(x − 2)y + 8y = 0 5. The following equations have a regular singular point at x = 0 and their indicial equations have two unequal roots that do not differ by an integer. For x > 0 find the first three non-zero terms in each of two linearly independent series solutions: a) 2xy + y + xy = 0 b) 2x2 y + 3xy + (2x2 − 1)y = 0 ...
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