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Assignment 1

# Assignment 1 - AM351 Spring 08 Due Monday June 9 Assignment...

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AM351 Spring 08 Due Monday, June 9 Assignment 1 Instructions: write your solutions clearly, explain your steps carefully and refer to the- orems and lemmas when needed. The assignment is due either in class or in the as- signments drop box (4th floor) by 2pm. 1. ( 3 marks ) Consider the following initial value problem involving Legendre’s equation (1 - x 2 ) y - 2 xy + α ( α + 1) y = 0 , y (0) = 1 , y (0) = 5 . On what interval does the existence and uniqueness theorem stated in class pre- dict that a unique solution to the problem will exist? Justify your answer. 2. ( 3 marks ) (a) Use the definition to show that the functions f 1 ( x ) = 1 , f 2 ( x ) = x, f 3 ( x ) = x 2 are linearly independent on the real line. (b) The above result can be generalized to show that the set of functions f 1 ( x ) = 1 , f 2 ( x ) = x, f 3 ( x ) = x 2 , ..., f n +1 ( x ) = x n are linearly independent on the real line. Use this result to prove that, for any constant r f 1 ( x ) = e rx , f 2 ( x ) = xe rx , f 3 ( x ) = x 2 e rx , ..., f n +1 ( x ) = x n e rx are linearly independent on the real line.

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Assignment 1 - AM351 Spring 08 Due Monday June 9 Assignment...

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