527ass3 - 642:527 ASSIGNMENT 3 FALL 2009 Turn in starred...

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642:527 ASSIGNMENT 3 FALL 2009 Turn in starred problems (including problem 3.A) on Tuesday 09/22/2009. DO NOT TURN IN SOLUTIONS FOR ANY UNSTARRED PROBLEMS. Multiple-page homework must be STAPLED when handed in. Section 4.3: 6. (e), (f), (u), *(v) Section 4.5 8 *9 Section 4.6 *2, *5(a) (see instructions in Remark 3 below) 6 16 *3.A As discussed in class the point x = 1 is a regular singular point for the Legendre equation (1 - x 2 ) y ′′ - 2 xy + λy = 0; the indicial equation is r 2 = 0 so there will be solutions ˜ y 1 ( x ) = n =0 a n ( x - 1) n and ˜ y 2 ( x ) = ˜ y 1 ( x ) ln | x - 1 | + n =1 b n ( x - 1) n . Find ˜ y 1 ( x ) and show that if λ = n ( n + 1) for some nonnegative integer n then ˜ y 1 ( x ) is a polynomial (you are not asked to ±nd ˜ y 2 ( x )). Remarks: 1. The problems from 4.2.6 illustrate the various possibilities for the Frobenius method. In Assignment 2 we solved 4.3.6(j), where r 1 - r 2 is not an integer; all the
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