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1- ordinary_steps

1- ordinary_steps - GE 207K Series Solutions Near an...

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Unformatted text preview: GE 207K Series Solutions Near an Ordinary Point March 3, 2011 This page aims to recap our mini-lecture on ‘series solution near an Ordinary point’. We wish expand the solutions of a differential equation that appears as P ( x ) y 00 + Q ( x ) y 00 + R ( x ) y = 0 , (1) about a point of interest, labeled hereon as x . Loosely, we call this point an ordinary point if P ( x ) 6 = 0 . For a more rigorous definition of an ordinary point, refer to Section 5.3 of your book. Problem statement: Find the series solution of the differential equation P ( x ) y 00 + Q ( x ) y + R ( x ) y = 0 , (2) about x . Steps: 1. Choose the appropriate form of solution y = ∞ X n =0 a n ( x- x ) n = a + a 1 ( x- x ) + a 2 ( x- x ) 2 + ··· (3) and take it’s derivatives ! I highly recommend that you move the starting point up by 1 everytime you differ- entiate. y = ∞ X n = 1 na n ( x- x ) n , y 00 = ∞ X n = 2 n ( n- 1) a n ( x- x ) n . 2. Plug y , y , y 00 back into ODE P ( x ) ∞ X n = 2 n ( n- 1) a n ( x- x ) n + Q ( x ) ∞ X n = 1 na n ( x- x ) n + R ( x ) ∞ X n =0 a n ( x- x ) n = 0 3. Move all the coefficients P ( x ) ,Q ( x ) , and R ( x ) into the summations. 4. Check the “lengths” of all series. That is, make the first term of all summations start from the same power of x . If the lengths are different, make them the same by doing one of the following: (i) Add zeroes to the shorter ∑ ’s (Always works in case of power series near an ordinary point). 1 GE 207K Series Solutions Near an Ordinary Point March 3, 2011 (ii) Take out terms from the larger ones. (Only option for singular points). 5. Shift indices to make all series start at the same n . Remember: X n ↓ n ↑ , X n ↑ n ↓ . (4) At this stage, if you’ve done everything right, the power of the series and the indices of all series should be the same....
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