4601HW2p2015 - Longman Homework No 2 EEME E4601 Problem 1...

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Longman’ Homework No. 2 EEME E4601 Problem 1 Considered’the’following’differential’equation d 2 y dt 2 + 3 dy dt + 2 y = 0 (A)’What’are’the’time’constants’in’the’solutions’to’this’homogeneous’equation? (B)’Say’what’we’defined’to’be’the’settling’time’of’a’s ystem.’ (C)’What’is’the’settling’time’of’this’system? Consider’the’following’difference’equation y ( k + 2) 5 4 y ( k + 1) + 3 8 y ( k ) = 0 (D)’What’are’the’roots’of’the’characteristic’polynomial?’What’are’the’two’linearly’ independent’solutions? ’ (E)’By’how’much’does’ea ch’solution’decay’each’time’step? (F)’If’the’samples’are’taken’at’ t = kT ,’and’ T ’is’1/10,’what’is’the’time’constant’ associated’with’each’solution?’What’is’the’settling’time’associated’with’each’ solution?’ (G)’How’do’the’answers’in’part’( F)’change’when’T’is’1/100? Problem 2 Find’particular’solutions’for’the’following’nonhomogeneous’differential’equations d 2 y dt 2 + 3 dy dt + 2 y = 5 d 2 y dt 2 + 3 dy dt + 2 y = t d 2 y dt 2 + 3 dy dt + 2 y = e t d 2 y dt 2 + 3 dy dt + 2 y = e t d 2 y dt 2 + 3 dy dt = 5 d 2 y dt 2 + 2 y = cos t
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Problem 3 The’handout’DFPartSol4601.pdf’has’a’table’of’“guesses”’to’substitute’into’a’ homogeneous’different ial’equation’in’order’to’find’a’particular’solution.’The’Table’ is’missing’the’“guess”’for’forcing’function f ( t ) = t cos( t ) .’ Following’the’logic’presented’in’class,’find’the’missing’entry.’There’is’no’guessing’ involved,’you’prove’that’the’guess’ will’work,’if’you’process’did’not’repeat’a’root’ that’was’there’already.
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