▁In ▁general , ▁the ▁variations ▁in ▁d amp ing ▁and ▁frequency ▁are ▁relatively ▁small ▁perturb ations ▁ ▁where ▁ ▁and ▁ ▁are ▁constants , ▁namely , ▁the ▁time - aver aged ▁oscill ator ▁frequency ▁and ▁d amp ing , ▁respectively . ▁ ▁The ▁transformed ▁frequency ▁can ▁be ▁written ▁in ▁a ▁similar ▁way : ▁, ▁ ▁where ▁ ▁is ▁the ▁natural ▁frequency ▁of ▁the ▁d amp ed ▁harm onic ▁oscill ator ▁ ▁and ▁. ▁ ▁Thus , ▁our ▁transformed ▁equation ▁can ▁be ▁written ▁. ▁ ▁The ▁independent ▁variations ▁ ▁and ▁ ▁in ▁the ▁oscill ator ▁d amp ing ▁and ▁reson ance ▁frequency , ▁respectively , ▁can ▁be ▁combined ▁into ▁a ▁single ▁p ump ing ▁function ▁. ▁The ▁con verse ▁conclusion ▁is ▁that ▁any ▁form ▁of ▁paramet ric ▁exc itation ▁can ▁be ▁accomplished ▁by ▁varying ▁either ▁the ▁reson ance ▁frequency ▁or ▁the ▁d amp ing , ▁or ▁both . ▁ ▁Solution ▁of ▁the ▁transformed ▁equation ▁Let ▁us ▁assume ▁that ▁ ▁is ▁sin us oid al , ▁specifically ▁ ▁where ▁the ▁p ump ing ▁frequency ▁ ▁but ▁need ▁not ▁equal ▁ ▁exactly . ▁The ▁solution ▁ ▁of ▁our ▁transformed ▁equation ▁may ▁be ▁written ▁ ▁where ▁the ▁rapidly ▁varying ▁components ▁have ▁been ▁fact ored ▁out ( ▁and ▁) ▁to ▁isol ate ▁the ▁slowly ▁varying ▁amplit udes ▁ ▁and ▁. ▁This ▁corresponds ▁to ▁Lap lace ' s ▁variation ▁of ▁parameters ▁method . ▁ ▁Sub st itut ing ▁this ▁solution ▁into ▁the ▁transformed ▁equation ▁and ▁retain ing ▁only ▁the ▁terms ▁first - order ▁in ▁ ▁yields ▁two ▁couple d