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Unformatted text preview: Exam 2, Version A Math 20D, Lecture C, Winter 2006 6 March 2006 Name: ID #: Section Time: Read all of the following information before starting the exam: Show all work, clearly and in order, if you want to get full credit. I reserve the right to take off points if I cannot see how you arrived at your answer (even if your final answer is correct). Justify your answers algebraically whenever possible to en- sure full credit. When you do use your calculator, sketch all relevant graphs and explain all relevant mathematics. Circle or otherwise indicate your final answers. Please keep your written answers brief; be clear and to the point. I will take points off for rambling and for incorrect or irrelevant statements. This test has 4 problems and is worth 100 points. It is your responsibility to make sure that you have all of the pages! Good luck! # Score 1 2a 2b 2c 2d 3a 3b 4a 4b Total 1. [ 25 points ] (Variation of Parameters) Given the two linearly independent solutions y 1 ( t ) = t, y 2 ( t ) = te t , to the linear, second-order homogeneous equation, t 2 y 00- t ( t + 2) y + ( t + 2) y = 0 , t > , determine the general solution to the non-homogeneous ordinary differential equation, t 2 y 00- t ( t + 2) y + ( t + 2) y = t 3 e 2 t , t > . Solution: In order to use the variation of parameters method, the differential equation must be in the form y 00 + p ( t ) y + q ( t ) y = g ( t ) . Since t > , we may divide every term by t 2 to convert the equation to the proper form, y 00- t + 2 t y + t + 2 t 2 y = te 2 t , Thus our non-homogeneous term is g ( t ) = te 2 t (5 pts for using the correct g ( t ) ). Given the two solutions y 1 ( t ) = t , y 2 ( t ) = te t , the Wronskian is given by (3 pts for correct Wronskian) W ( y 1 , y 2 )( t ) = y 1 y 2- y 1 y 2 = te t + t 2 e t- te t = t 2 e t . We now have all of the necessary components to use the variation of parameters formula for the particular solution Y ( t ) (7 pts for correct setup, 5 points for correct Y ( t ) ): Y ( t ) =- y 1 ( t ) Z y 2 ( t ) g ( t ) W ( y 1 , y 2 )( t ) dt + y 2 ( t ) Z y 1 ( t ) g ( t ) W ( y 1 , y 2 )( t ) dt (1) =- t Z te t ( te 2 t )...
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This note was uploaded on 06/12/2008 for the course MATH 20D taught by Professor Mohanty during the Spring '06 term at UCSD.
- Spring '06