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Homework 6 solutions

# Homework 6 solutions - AMS 361 Applied Calculus IV by Prof...

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AMS 361: Applied Calculus IV by Prof Y. Deng Homework 6 Assignment Date: Wednesday (03/18/200 9 ) Collection Date: Wednesday (04/01/2009) No HW Due 3/25 (Quiz2 Date) Grade: Each problem is worth 10 points. Problem 6.1 Find the general solution of the following equation ሺܦ െ 4ሻ ሺܦ െ 2ሻ ሺܦ െ 2ܦ െ 3ሻ ൌ 0 where ܦ ؠ ݀ ݀ݔ Solution: The characteristic equation is . 0 ) 3 2 ( ) 2 ( ) 4 ( 2 2 4 = r r r r We have . 1 , 3 , 2 , 4 8 7 6 5 4 3 2 1 = = = = = = = = r r r r r r r r Therefore the general solution is . ) ( ) ( ) ( 8 3 7 2 6 5 4 3 4 2 3 2 1 x x x x e c e c e x c c e x c x c x c c x y + + + + + + + = Problem 6.2 Find the general solution of the following equation ൬ݔ ݀ ݀ݔ െ ߙ൰ ݕሺݔሻ ൌ 0 where ߙ ൌ constant and ݊ ൌ integer Solution: Let , ln x r = ), ( ) ( r e y r u = we have dr d x dx dr dr d dx d 1 = = , . 1 α α α = = dr d dr d x x dx d x . 0 ) ( ) ( ) ( ) ( ) ( ) ( = = = r u dr d e y dr d x y dx d x n r n n α α α Then . ... 2 1 α = = = = n r r r We have . ) ... ( ) ( 1 1 2 1 r n n e r c r c c r u α + + + = . ) ... ( ) ( 1 1 2 1 r n n r e r c r c c e y α + + + = Then the general solution is { . } ) (ln ... ln ) ( 1 1 2 1 α x x c x c c x y n n + + + = Problem 6.3 (Prob. 34, P. 168 Modified) Compute the following Vandermonde

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