▁used ▁to ▁model ▁d amp ed ▁oscill ators . ▁ ▁The ▁equation ▁is ▁given ▁by ▁▁▁▁ ▁where ▁ ƒ ( y ) ▁is ▁a ▁piece wise ▁constant ▁function ▁which ▁is ▁positive , ▁except ▁for ▁small ▁y ▁as ▁▁▁▁▁▁▁ ▁Eq . ▁( 1 ) ▁is ▁a ▁special ▁case ▁of ▁the ▁L ien ard ▁equation ; ▁it ▁describes ▁the ▁auto - osc ill ations . ▁ ▁Solution ▁( 1 ) ▁at ▁a ▁separate ▁time ▁intervals ▁when ▁f ( y ) ▁is ▁constant ▁is ▁given ▁by ▁▁▁▁ ▁Here ▁ ▁, ▁ ▁at ▁▁ ▁and ▁▁ ▁otherwise . ▁Expression ▁( 2 ) ▁can ▁be ▁used ▁for ▁real ▁and ▁complex ▁values ▁of ▁. ▁ ▁The ▁first ▁half - period ’ s ▁solution ▁at ▁ ▁is ▁▁▁▁▁▁▁▁▁▁▁ ▁The ▁second ▁half - period ’ s ▁solution ▁is ▁▁▁▁ ▁The ▁solution ▁contains ▁four ▁constants ▁of ▁integration ▁, ▁, ▁, ▁, ▁the ▁period ▁ ▁and ▁the ▁boundary ▁ ▁between ▁ ▁and ▁ ▁needs ▁to ▁be ▁found . ▁A ▁boundary ▁condition ▁is ▁derived ▁from ▁continu ity ▁of ▁) ▁and ▁. ▁ ▁Solution ▁of ▁( 1 ) ▁in ▁the ▁station ary ▁mode ▁thus ▁is ▁obtained ▁by ▁solving ▁a ▁system ▁of ▁algebraic ▁equations ▁as ▁ ▁; ▁; ▁; ▁; ▁; . ▁ ▁The ▁integration ▁constants ▁are ▁obtained ▁by ▁the ▁Le ven berg – Mar qu ard t ▁algorithm . ▁ ▁With ▁, ▁, ▁ ▁Eq . ▁( 1 ) ▁named ▁Van ▁der ▁Pol ▁oscill ator . ▁Its ▁solution ▁cannot ▁be ▁expressed ▁by ▁elementary ▁functions ▁in ▁closed ▁form . ▁ ▁References ▁ ▁Category : D iffer