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Unformatted text preview: AMATH 351 Assignment No. 4 Fall 2005 Questions 1-5(a) are due on Thursday, October 20, 2005, 1:30 p.m.. (You can slide the assignment under my door if I am not in my office.) 1. In the Course Notes, p. 7-9, it is shown that any solution of the DE x 2 d 2 y dx 2 + (1 + 2 c ) x dy dx + ( a 2 x 2 b + c 2 b 2 p 2 ) y = 0 , (1) where a , b and c are constants, is of the form y ( x ) = x c w parenleftBig a b x b parenrightBig , (2) where w ( z ) is a solution of Bessels DE z 2 d 2 w dz 2 + z dw dz + ( z 2 p 2 ) w = 0 . (3) (a) Use the above result (you dont have to derive it!) to find the general solution of the DE xy + 2 y + xy = 0 (4) in terms of Bessel functions. (This question is from the Course Notes, Problem Set 2, Q. 3.) (b) Show that the general solution to this DE can also be expressed in terms of trigonometric functions. (Hint: Look at the value of p .) (c) Given any A, B R , does there exist a unique solution to the initial value problem y (0) = A , y (0) = B ?...
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This note was uploaded on 09/20/2010 for the course AMATH 351 taught by Professor Sivabalsivaloganathan during the Spring '08 term at Waterloo.
- Spring '08