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Part A (. (True/False Questions). Choose True or False. 1) The Laplace transform of ๐‘ฆโ€ฒโ€ฒโ€ฒ is ๐‘ !๐‘Œ ๐‘  โˆ’ ๐‘ !๐‘ฆ 0 โˆ’ ๐‘ ๐‘ฆ! 0 โˆ’ ๐‘ฆ 0 . โ–ก True โ–ก False 2) The Laplace transform of ๐‘’!!! ! is ! !!!! . โ–ก True โ–ก False 3) The inverse Laplace transform of ๐น ๐‘  = (!!)! !!!!! is ๐‘ก!. โ–ก True โ–ก False 4) โ„’{4 + 2t โˆ’ 5 cos(4๐‘ก)} is equal to โ„’{4} +2โ„’{ ๐‘ก}โˆ’5โ„’{cos(4๐‘ก)}. โ–ก True โ–ก False 5) โ„’!!{! ! + ! !!!!} is equal to 3 + โ„’!!{ ! !!!!}. โ–ก True โ–ก False Part B (Multiple Choice Questions). Chose the correct answer. 1) The Laplace transform of sin(! !) is a) !! !!!!! b) ! !!!!! c) ๐Ÿ ๐Ÿ’๐’”๐Ÿ!๐Ÿ d) ! !!!!! e) None of them 2) The Laplace transform of ๐‘’!! โˆ’ ๐‘’!!! is a) ๐Ÿ‘๐’” ๐’”๐Ÿ!๐Ÿ b) !๐Ÿ”๐’” ๐’”๐Ÿ!๐Ÿ c) !๐Ÿ ๐’”๐Ÿ!๐Ÿ d) None of them e) ๐Ÿ” ๐’”๐Ÿ!๐Ÿ— 3) The inverse Laplace transform of ๐น ๐‘  = ! !!! + ! !! is a) 4๐‘’!! + 4๐‘ก b) 4๐‘’! + 4๐‘ก c) 4๐‘’!! โˆ’ 4๐‘ก d) 4๐‘’! โˆ’ 4๐‘ก e) None of them 4) The inverse Laplace transform of ๐น ๐‘  = ! !!!! โˆ’ !! !!!! is a) 2sin ๐‘ก โˆ’ 3 cos(3๐‘ก) b) sin(2๐‘ก) โˆ’ 3 cos(3๐‘ก) c) sin(2๐‘ก) โˆ’ cos( 3๐‘ก) d) None of them e) sin(2๐‘ก) + 3 cos( 3๐‘ก) 5) When the Laplace transform is applied to the problem y! + ๐‘ฆ = 0, y 0 = 1, the resulting transformed equation is a) s + 1 Y s = 1 b) ๐‘  + 1 ๐‘Œ ๐‘  = โˆ’! ! c) ๐‘  + 1 ๐‘Œ ๐‘  = 0 d) ๐‘  + 1 ๐‘Œ ๐‘  = ! ! e) None of them Part C . (Workout Problems). Solve the following problems. Show your work clearly. 1) a) Evaluate โ„’{3๐‘ก!} by using definition of the Laplace transform. b) Can you use the result in part a) to evaluate โ„’{3๐‘ก!}? (Why? Why not?) 2) Given initial value problem y!!! = f t , y 0 = 0, y! 0 = 0, y!! 0 = 4. a) Let f(t) = 0 and use the Laplace transform to solve the above initial value problem. b) Let f t = 3 and use the Laplace transform to solve the above initial value problem. 3) a) Find the inverse Laplace transform of ๐‘Œ ๐‘  = ! (!!!)(!!!!) + ! !!! . b) Use the Laplace transform to solve the initial value problem ๐‘ฆ! โˆ’ 3๐‘ฆ = cos ๐‘ก, y 0 = 2. ๐œ) Do you see a relationship between part a) and part b)?

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Engineering: Laplace Transform Questions
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