## 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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