Exercise Set 4

# Exercise Set 4 - Exercise Set 4 Math 4027 Due: April 7,...

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Math 4027 Due: April 7, 2005 1. Find the general solution of each of the following diﬀerential equations. (a) 2 x 2 y 00 + xy 0 - y = 0 I Solution. The indicial equation q ( r ) = 2 r ( r - 1) + r - 1 = (2 r + 1)( r - 1) has roots 1 and - 1 / 2. Hence y = c 1 x + c 2 | x | - 1 / 2 . J (b) 2 x 2 y 00 - 3 xy 0 + 2 y = 0 I Solution. The indicial equation q ( r ) = 2 r ( r - 1) - 3 r + 2 = 2 r 2 - 5 r + 2 = (2 r - 1)( r - 2) has roots 2 and 1 / 2. Hence y = c 1 x 2 + c 2 | x | 1 / 2 . J (c) 9 x 2 y 00 + 2 y = 0 I Solution. The indicial equation q ( r ) = 9 r ( r - 1) + 2 = 9 r 2 - 9 r + 2 = (3 r - 1)(3 r - 2) has roots 1 / 3 and 2 / 3. Hence y = c 1 | x | 1 / 3 + c 2 | x | 2 / 3 . J (d) x 2 y 00 - 3 xy 0 + 4 y = 0 I Solution. The indicial equation q ( r ) = r ( r - 1) - 3 r +4 = r 2 - 4 r +4 = ( r - 2) 2 has a single root 2 of multiplicity 2. Hence y = c 1 x 2 + c 2 x 2 log | x | . J (e) x 2 y 00 - 5 xy 0 + 9 y = 0 I Solution. The indicial equation q ( r ) = r ( r - 1) - 5 r +9 = r 2 - 6 r +9 = ( r - 3) 2 has a single root 3 of multiplicity 2. Hence

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## This note was uploaded on 03/20/2011 for the course MATH 4027 taught by Professor Adkins during the Spring '06 term at LSU.

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Exercise Set 4 - Exercise Set 4 Math 4027 Due: April 7,...

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