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Unformatted text preview: Engineering Analysis 4 Workshop questions - week of 10/26/08 1. Consider the forced oscillator x 00 + x = cos 2 t. (1) (a) Find the general solution. Answer- this is a straightforward review of undamped oscillators. The natural frequency is 1 and x c = cos t + sin t Now you have to find a particular solution. Take as a trial solution x p ( t ) = A cos 2 t + B sin 2 t, and plug this into equation (1). You get- 3 A cos 2 t- 3 B sin 2 t = cos 2 t. (2) Equating coefficients of cos 2 t and sin 2 t in equation (2) you can see that A =- 1 / 3 and B = 0 . Thus, the particular solution is x p ( t ) =- 1 3 cos 2 t and the general solution is x = cos t + sin t- 1 3 cos 2 t. (b) Suppose you are interested in the initial conditions x (0) = 1 , x (0) = 0 . Explain the fallacy in the following argument. Incorrect argument - determine and to match the initial conditions so that 1 = cos 0 + sin 0 , 0 =- sin 0 + cos 0 . or = 1 , = 0 . Thus, the solution to the initial value problem is x ( t ) = cos t- 1 3 cos 2 t, which clearly does not satisfy the initial condition. Answer - Facilitators please note - this is a very common source of error on exams. Please make certain that the students clearly understand what is wrong with this reasoning....
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This note was uploaded on 09/08/2010 for the course GEN_ENG 205 taught by Professor Taflove during the Spring '08 term at Northwestern.
- Spring '08