Behavior of Solutions as t . 37 through 39 are concerned

Chapter 3, Problem 39

(choose chapter or problem)

Behavior of Solutions as t . 37 through 39 are concerned with the behavior of solutions as t .Show that y = sin t is a solution ofy+ (k sin2 t)y+ (1 k cost sin t)y = 0for any value of the constant k. If 0 < k < 2, show that 1 k cost sin t > 0 and k sin2 t 0.Thus observe that even though the coefficients of this variable-coefficient differential equationare nonnegative (and the coefficient of yis zero only at the points t = 0, , 2, . . .),it has a solution that does not approach zero as t . Compare this situation with theresult of 37. Thus we observe a not unusual situation in the study of differentialequations: equations that are apparently very similar can have quite different properties.