Homework-7-f11-solution

Homework-7-f11-solution - 1 AMS 361: Applied Calculus IV...

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Unformatted text preview: 1 AMS 361: Applied Calculus IV (DE & BVP) Homework 7 Assignment Date: Thursday (11/03/2011) Collection Date: Thursday (11/10/2011) Grade: Each problem is worth 10 points Problem 7.1: Find the particular solution for ࠵ !!! + ࠵ !! + ࠵ ! + ࠵ = 1 + ࠵ + ࠵ ! + sin ࠵ Solution: First, we need to find the roots of the characteristic equation, ࠵ ! + ࠵ ! + ࠵ + 1 = ࠵ ! = − 1 , ࠵ ! = ࠵ , ࠵ ! = − ࠵ We get the following common solution. ࠵ ! = ࠵ ࠵ ! ! + ࠵ sin ࠵ + ࠵ cos ࠵ From the common solution, we can deduce the following particular solution. ࠵ ! = ࠵ ! + ࠵ ! ࠵ + ࠵ ! ࠵ ! + ࠵ ( ࠵ ! ࠵¡¢ ࠵ + ࠵ ! ࠵¡¢ ࠵ ) Then ࠵ ! ! = ࠵ ! + ࠵ ! ࠵ ! + ࠵ ! cos ࠵ + ࠵ ! sin ࠵ + ࠵ ( − ࠵ ! sin ࠵ + ࠵ ! cos ࠵ ) ࠵ ! !! = ࠵ ! ࠵ ! − 2 ࠵ ! sin ࠵ + 2 ࠵ ! cos ࠵ + ࠵ ( − ࠵ ! cos ࠵ − ࠵ ! sin ࠵ ) ࠵ ! !!! = ࠵ ! ࠵ ! − 3 ࠵ ! cos ࠵ − 3 ࠵ ! sin ࠵ + ࠵ ( ࠵ ! ࠵¡¢ ࠵ − ࠵ ! cos ࠵ ) Through substitution of these results into original D.E., we get A ! + ࠵ ! + ࠵ ! ࠵ + 4 ࠵ ! ࠵ ! + − 2 ࠵ ! + 2 ࠵ ! cos ࠵ + − 2 ࠵ ! − 2 ࠵ ! sin ࠵ = 1 + ࠵ + ࠵ ! + sin ࠵ By comparing both sides, we get 2 ࠵ ! + ࠵ ! = 1 ࠵ ! = 1 4 ࠵ ! = 1 − 2 ࠵ ! + 2 ࠵ ! = − 2 ࠵ ! − 2 ࠵ ! = 1 From this system of equations, we have ࠵ ! = ࠵ ! = 1 ࠵ ! = 1 4 ࠵ ! = − 1 4 ࠵ ! = − 1 4 Finally, the particular solution is ࠵ !...
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This note was uploaded on 11/14/2011 for the course AMS 361 taught by Professor Staff during the Fall '08 term at SUNY Stony Brook.

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Homework-7-f11-solution - 1 AMS 361: Applied Calculus IV...

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