Lecture 4 Notes

# Independence and the wronskian section 32 two

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Unformatted text preview: ey aren’t solutions to an ODE. W (y1 , y2 )(t) = ￿ y1 (t)y2 (t) − ￿ y1 (t)y2 (t) Independence and the Wronskian (Section 3.2) • Two functions y1(t) and y2(t) are linearly independent provided that the only way that C1y1(t) + C2y2(t) = 0 for all values of t is when C1=C2=0. e.g. y1 (t) = e2t+3 and y2 (t) = e2t−3 are not independent. • The Wronskian is deﬁned for any two functions, even if they aren’t solutions to an ODE. W (y1 , y2 )(t) = ￿ y1 (t)y2 (t) − ￿ y1 (t)y2 (t) • If the Wronskian is nonzero for some t, the functions are linearly independent. Independence and the Wronskian (Section 3.2) • Two functions y1(t) and y2(t) are linearly independent provided that the only way that C1y1(t) + C2y2(t) = 0 for all values of t is when C1=C2=0. e.g. y1 (t) = e2t+3 and y2 (t) = e2t−3 are not independent. • The Wronskian is deﬁned for any two functions, even if they aren’t solutions to an ODE. W (y1 , y2 )(t) = ￿ y1 (t)y2 (t) − ￿ y1 (t)y2 (t) • If the Wronskian is nonzero for some t, the functions are linearly independent. • If y1(t) and y2(t) are solutions to an ODE and the Wronskian is nonzero then they are independent and y (t) = C1 y1 (t) + C2 y2 (t) is the general solution. We call y1(t) and y2(t) a fundamental set of solutions. Independence and the Wronskian (Section 3.2) • So for case i (distinct roots), can we form a general solution from y1 (t) = er1 t and y2 (t) = er2 t ? Independence and the Wronskian (Section 3.2) • So for case i (distinct roots), can we form a general solution from y1 (t) = er1 t • Must check the Wronskian: and y2 (t) = er2 t ? Independence and the Wronskian (Section 3.2) • So for case i (distinct roots), can we form a general solution from y1 (t) = er1 t y2 (t) = er2 t ? and • Must check the Wronskian: W (e r1 t r2 t ,e )(t) = e r1 t r2 t r2 e r1 t r 2 t − r1 e e Independence and the Wronskian (Section 3.2) • So for case i (distinct roots), can we form a general solution from y1 (t) = er1 t y2 (t) = er2 t ? and • Must check the Wronskian: W (e r1 t r2 t ,e )(t) = e r1 t r2 t r1 t r 2 t − r1 e r2 e = (r1 − r2 )e r...
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