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Unformatted text preview: 1 AMS 361: Applied Calculus IV by Prof Y. Deng Homework 6 Assignment Date: Wednesday (02/29/2012) Collection Date: Wednesday (03/21/2012) Grade: Each problem is worth 10 points. Problem 6.1 Find the general solution of the following equation 3 ! 2 ! ! + 2 ( ) = where Solution: From the DE, general solution is given by sum of general solutions of the three DEs 3 ! = 2 ! = ! + 2 = For 3 ! = , solution is = ! + ! ! ! For 2 ! = , solution is = ! + ! + ! ! ! ! For ! + 2 = , solution is = ! ! ! + ! ! ! , where ! , ! are zeros of the characteristic polynomial ! + 2 = ! , ! = ! ! ! ! ! If we like to express the solution in real-value (at least it appears), it can be rewritten as = ! ! ! sin ! ! + ! cos ! ! The general solution to the original DE is then = ! + ! ! ! + ! + ! + ! ! ! ! + ! ! ! sin 7 2 + ! cos 7 2 . Problem 6.2 Find the general solution of the following equation ! = where = constant and = positive integer 2 Solution: The solution to ! !" ! = has multiplicity , so the general solution will contain terms similar but linearly independent from the first order equation....
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