Apuendix A Solrrl ions to the Exerrises 483 Putting in YI and 15, y r ( t )-= ~7('~-') +c ~('lt-0) )liltion: = cos(4t - (3) is oW,ained g ( f ) = y)L(t)r i/,(t) lhtcrrnining c from the initial condition., + y(0) = yo = 1/fL(0) U , ( [ ) ) =- c + 'Lcos(0) \ -4/.-d =12 c = yo 1.2 - Now t hc cornplete isolution can be given: y(t) + (go F + = ' -pd internal part external part ~ o l ~ t i o7.3 Ii it) H ( s )= s b) X(h) +3 s 10 j RJ 16 Y ( s ) = H ( s ) X(") 7[vo - 01 5a3 10 = + s f + S2+1li + _ s + [MO - 01 (using Eq 7.16) r) F'wtial fraction clec*oinposition: A l3 B* I - l ( S ) X ( S )= -+ -$ S f S s 71 S - J4 -3.10 A = [ ( s ) H j s ) X ( s ) ] / = - -1.2 51-3 25 + Iriwrse transfor uratiorr: Y(t) = P { H ( s ) X ( s ) } L - l { + }.- 5t-3 ] f ( t ) + 9" e-dt Conversion io cosiires H and B* in rrragnitiitlc and pliase * €(I)