Chapter 4Fort= 0,y1(0) = 3·02= 0 andy2(0) = 0. The latter follows from the fact that one-sidedderivatives ofy2(t), 3t2and−3t2, are both zero att= 0. Also,y1(0) =y2(0) = 0. HenceW[y1, y2](0) =0000= 0,and soW[y1, y2](t)≡0 on (−∞,∞). This result does not contradict part (b) in Prob-lem 34 because these functions are not a pair of solutions to a homogeneous linearequation with constant coeﬃcients.37.Ify1(t) andy2(t) are solutions to the equationay+by+c= 0, then, by Abel’s formula,W[y1, y2](t) =Ce−bt/a, whereCis a constant depending ony1andy2. Thus, ifC= 0, thenW[y1, y2](t) = 0 for anytin (−∞,∞), because the exponential function,e−bt/a, is never zero.ForC= 0,W[y1, y2](t)≡0 on (−∞,∞).39.(a)A linear combination ofy1(t) = 1,y2(t) =t, andy3(t) =t2,C1·1 +C2·t+C3·t2=C1+C2t+C3t2,is a polynomial of degree at most two and so can have at most two real roots, unless it isa zero polynomial, i.e., has all zero coeﬃcients. Therefore, the above linear combinationvanishes on (
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