# D find a basis for the space of solutions to the

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Unformatted text preview: the initial conditions. Use the recursion formula to determine an in terms of a0 and a1 , for 2 ≤ n ≤ 9. c. Find a nonzero polynomial solution to this diﬀerential equation. 13 d. Find a basis for the space of solutions to the equation. e. Find the solution to the initial value problem d2 y dy − 4x + 12y = 0, dx2 dx dy (0) = 1. dx y (0) = 0, f. To solve the diﬀerential equation d2 y dy − 4(x − 3) + 12y = 0, 2 dx dx it would be most natural to assume that the solution has the form ∞ an (x − 3)n . y= n=0 Use this idea to ﬁnd a polynomial solution to the diﬀerential equation d2 y dy − 4(x − 3) + 12y = 0. dx2 dx 1.2.2. We want to use the power series method to ﬁnd the general solution to Legendre’s diﬀerential equation (1 − x2 ) d2 y dy − 2x + p(p + 1)y = 0. dx2 dx Once again our approach is to assume our solution is a power series centered at 0 and determine the coeﬃcients in this power series. a. As a ﬁrst step, ﬁnd the recursion formula for an+2 in terms of an . b. Use the recursion formula to determine an in terms of a0 and a1 , for 2 ≤ n ≤ 9. c. Fin...
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## This document was uploaded on 01/12/2014.

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