# Engineering: Laplace Transform Questions

## Transcribed Text

Part A (. (True/False Questions). Choose True or False. 1) The Laplace transform of 𝑦′′′ is 𝑠!𝑌 𝑠 − 𝑠!𝑦 0 − 𝑠𝑦! 0 − 𝑦 0 . □ True □ False 2) The Laplace transform of 𝑒!!! ! is ! !!!! . □ True □ False 3) The inverse Laplace transform of 𝐹 𝑠 = (!!)! !!!!! is 𝑡!. □ True □ False 4) ℒ{4 + 2t − 5 cos(4𝑡)} is equal to ℒ{4} +2ℒ{ 𝑡}−5ℒ{cos(4𝑡)}. □ True □ False 5) ℒ!!{! ! + ! !!!!} is equal to 3 + ℒ!!{ ! !!!!}. □ True □ False Part B (Multiple Choice Questions). Chose the correct answer. 1) The Laplace transform of sin(! !) is a) !! !!!!! b) ! !!!!! c) 𝟐 𝟒𝒔𝟐!𝟏 d) ! !!!!! e) None of them 2) The Laplace transform of 𝑒!! − 𝑒!!! is a) 𝟑𝒔 𝒔𝟐!𝟏 b) !𝟔𝒔 𝒔𝟐!𝟏 c) !𝟐 𝒔𝟐!𝟏 d) None of them e) 𝟔 𝒔𝟐!𝟗 3) The inverse Laplace transform of 𝐹 𝑠 = ! !!! + ! !! is a) 4𝑒!! + 4𝑡 b) 4𝑒! + 4𝑡 c) 4𝑒!! − 4𝑡 d) 4𝑒! − 4𝑡 e) None of them 4) The inverse Laplace transform of 𝐹 𝑠 = ! !!!! − !! !!!! is a) 2sin 𝑡 − 3 cos(3𝑡) b) sin(2𝑡) − 3 cos(3𝑡) c) sin(2𝑡) − cos( 3𝑡) d) None of them e) sin(2𝑡) + 3 cos( 3𝑡) 5) When the Laplace transform is applied to the problem y! + 𝑦 = 0, y 0 = 1, the resulting transformed equation is a) s + 1 Y s = 1 b) 𝑠 + 1 𝑌 𝑠 = −! ! c) 𝑠 + 1 𝑌 𝑠 = 0 d) 𝑠 + 1 𝑌 𝑠 = ! ! e) None of them Part C . (Workout Problems). Solve the following problems. Show your work clearly. 1) a) Evaluate ℒ{3𝑡!} by using definition of the Laplace transform. b) Can you use the result in part a) to evaluate ℒ{3𝑡!}? (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 𝑌 𝑠 = ! (!!!)(!!!!) + ! !!! . b) Use the Laplace transform to solve the initial value problem 𝑦! − 3𝑦 = cos 𝑡, y 0 = 2. 𝐜) Do you see a relationship between part a) and part b)?

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