Part A (15 pts). (True/False Questions). Choose true or false. 1) ...

Part A (15 pts). (True/False Questions). Choose true or false. 1) The Laplace transform of cos 3𝑡 is ! !!!!! □ True □ False 2) The Laplace transform of 𝑒!!"! is ! !!!". □ True □ False 3) ℒ{3 + 4 t + 5 sin 2𝑡} is equal to 3ℒ{1} +4 ℒ{ 𝑡}+5ℒ{sin 2𝑡}. □ True □ False 4) ℒ!!{ !" !!!!} is equal to ℒ!!{15} ℒ!!{ ! !!!!}. □ True □ False 5) The inverse Laplace transform of 𝐹 𝑠 = !! !! is 𝑡!. □ True □ False Part B (25 pts). (Multiple Choice Questions). Chose the correct answer. 1) The Laplace transform of sin 4𝑡 is a) 𝟒 𝒔𝟐!𝟒𝟐 b) !! !!!!! c) !! !!!!! d) ! !!!!! e) None of them 2) The Laplace transform of 4 − 5 sin 𝑡 + 6 𝑐𝑜𝑠𝑡 − 7𝑒! is is Type equation here. a) 𝟒 𝒔 + 𝟓 𝒔𝟐!𝟏 + ! 𝒔 𝒔𝟐!𝟏 − ! 𝒔!𝟏 b) 𝟒 𝒔 − 𝟓 𝒔𝟐!𝟏 − ! 𝒔 𝒔𝟐!𝟏 − ! 𝒔!𝟏 c) 𝟒 𝒔 − 𝟓 𝒔𝟐!𝟏 + 𝟔𝒔 𝒔𝟐!𝟏 + ! 𝒔!𝟏 d) None of them e) 𝟒 𝒔 − 𝟓 𝒔𝟐!𝟏 + ! 𝒔 𝒔𝟐!𝟏 − ! 𝒔!𝟏 3) The inverse Laplace transform of 𝐹 𝑠 = ! !!! + ! !!!! is a) 𝑒!! − 5 𝑠𝑖𝑛𝑡 b) 𝑒!!! + 5 𝑐𝑜𝑠 𝑡 c) 𝑒!! + 5 𝑠𝑖𝑛𝑡 d) 𝑒!! + 5 𝑐𝑜𝑠 𝑡 e) None of them 4) The inverse Laplace transform of 𝐹 𝑠 = ! (!!!!) + !! (!!!!) is a) 2sin 𝑡 + 4 cos 𝑡 b) 2sin 2𝑡 + 4 cos 3𝑡 c) 2sin 2𝑡 − 4 cos 3𝑡 d) 4sin 2𝑡 + 12 cos 3𝑡 e) None of them 5) When the Laplace transform is applied to the problem y! + 5𝑦 = 1, y 0 = 0, the resulting transformed equation is a) 𝑠 + 5 𝑌 𝑠 = ! !! b) 𝑠 − 5 𝑌 𝑠 = ! ! c) 𝑠 + 5 𝑌 𝑠 = −! ! d) s + 5 Y s = ! ! e) None of them Part C (60 pts). (Workout Problems). Solve the following problems. Show your work clearly. 1) a) Find the Laplace transform of 𝑒! + 𝑒!! ! + sin 2𝑡 + 3 b) Find the inverse Laplace transform of 𝐹 𝑠 = ! !(!!!!). 2) Given initial value problem 𝑦! + 5𝑦 = 𝑒!!, y 0 = 1 a) Use the Laplace transform to solve the above initial value problem b) How do you solve the above initial value problem if we replace y 0 = 1 with y 1 = 1. (Justify your answer) 3) Given initial value problem 𝑦!! − 5y! + 6𝑦 = 𝑒!! , y 0 = 0, y! 0 = 1 a) Use the Laplace transform to solve the above initial value problem b) Do you know another method to solve the above initial value problem besides the Laplace transform? (Justify your answer)