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ass4 - AMATH 351 Assignment No 4 Fall 2005 Questions 1-5(a...

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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 Bessel’s DE z 2 d 2 w dz 2 + z dw dz + ( z 2 p 2 ) w = 0 . (3) (a) Use the above result (you don’t 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 ? i. If the answer is “yes”, find this solution. ii. If the answer is “no”, explain why a solution cannot be found. What condition(s) can you place on the general solution in order to isolate a specific solution?
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