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Winter 2006 - Reynold's Class - Quiz 3 (Version B)

Winter 2006 - Reynold's Class - Quiz 3 (Version B) - Math...

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Math 20D, Lecture C, Winter 2006 17 February 2006 Quiz 3, version B Name: ID #: Section Time: Show all work clearly and in order, and circle your final answers. You have 20 minutes to take this 25 point quiz. 1. [ 9 points ] Solve the given initial value problem: y - 4 y + 4 y = 0 , y (0) = 0 , y (0) = 2 Solution: The homogeneous second-order equation has characteristic equation r 2 - 4 r + 4 = 0 , which has the double-root at r = 2 (3 pts). Therefore, the two linearly-independent solutions to the equation are y 1 ( t ) = e 2 t , y 2 ( t ) = te 2 t , and the general solution to the equation is given by (3 pts) y ( t ) = c 1 e 2 t + c 2 te 2 t . We then find the constants c 1 and c 2 so that the solution satisfies the initial conditions, i.e. 0 = y (0) = c 1 2 = y (0) = 2 c 1 + c 2 , which has the unique solution at c 1 = 0 , c 2 = 2 . Thus the solution to the initial value problem is (3 pts) y ( t ) = 2 te 2 t .
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2. [ 8 points ] Determine a suitable form for the particular solution to Y ( t ) if the Method of Undetermined Coefficients is to be used on the following problem ( Do not solve the differential equation ): y - 6 y + 9 y = t sin(2 t ) + e 5 t Solution: We must use enough coefficients to handle the linear term in
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