MA 36600 LECTURE NOTES: FRIDAY, FEBRUARY 20Linear Homogeneous EquationsExample.Consider the differential equationy+ 5y+ 6y= 0.We discuss the Wronskian associated with this equation. We have the characteristic equationr2+ 5r+ 6 = 0.Since the polynomial factors asr2= 5r+6 = (r+ 2) (r+ 3), the characteristic equation has the two distinctreal rootsr1=−2 andr2=−3. Hence two solutions to the differential equation arey1(t) =e−2tandy2(t) =e−3t.The Wronskian ofy1andy2is the functionW(t) =y1y2−y1y2=e−2t−3e−3t−−2e−2te−3t=−3e−5t+ 2e−5t=−e−5t.Example.Consider the more general constant coeﬃcient equationa y+b y+c y= 0.We have the characteristic equationa r2+b r+c= 0.Say that this has rootsr1andr2, and consider the solutionsy1(t) =er1tandy2(t) =er2t.The Wronskian ofy1andy2is the functionW(t) =y1y2−y1y2=er1tr2er2t−r1er1ter2t= (r2−r1)e(r1+r2)t.
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