2

QI Answerthefollowingquestions:

1) Determine the largest interval !on which there exists a unique solution to the

initial value problem. Explain in details [2mark]! + 1 !!! − cos ! !! + 3! = 1; ! 0 = 1 , !! 0 = 0

2) The two-parameter family ! = !! cos5 ! + !! sin 5! is a solution of the differential equation !!! + 25! = 0. Determine whether a member of the family can be found that satisfies the boundary conditions. [1mark]

! 0 = 1, ! !5 = 1

3

3) !! ! = sin 3!isasolutionof!!! + 9! = 0.Usereductionofordertofindthe

secondsolution!!(!). [3marks]

4) Findlinearlyindependentfunctionsthatareannihilatedbythegivendifferentialoperator. [2marks]

(a) (! − 6)(2! + 3)

(b) !! − 6! + 10

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QII Solvethefollowingdifferentialequations1) !!! − 2!! + ! = !!!.(Usethemethodofundeterminedcoefficients)

[4marks]

5

2) 3!!! − 6!! + 6! = !! sec ! [4marks]

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QIII Solvethefollowinginitialvalueproblem [4marks]

!!!!! + !!! + ! = 0; ! 1 = 1 , !! 1 = 2

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QIV [5Marks]Answerthefollowingquestions:

1) ℒ !! =

2) ℒ {sin 4!} =

3) ℒ {cosh 2!} =

4) ℒ {!!!! cos 3!} =

5) ℒ {! cos !} =