engineertobe
New member
- Joined
- Oct 8, 2011
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Let y1 and y2 be two solutions of A(x)y" + B(x)y' + C(x)y = 0 on an open interval I where A, B, and C are continuous and A(x) is never = 0.
(a) Let W = (y1, y2)
Show that A(x) (dW/dx) = (y1)(Ay2")-(y2)(Ay1")
Then substitute for Ay2" and Ay1" from the original differential equation to show that A(x)(dW/dx) = -B(x)W(x)
(b) Solve this equation to deduce W(x) = K exp (- int( B(x) / A(x) )dx) where K is a constant.
(c) Why does this formula imply that the Wornskian W(y1, y2) is either zero everywhere or nonzero everywhere?
(a) Let W = (y1, y2)
Show that A(x) (dW/dx) = (y1)(Ay2")-(y2)(Ay1")
Then substitute for Ay2" and Ay1" from the original differential equation to show that A(x)(dW/dx) = -B(x)W(x)
(b) Solve this equation to deduce W(x) = K exp (- int( B(x) / A(x) )dx) where K is a constant.
(c) Why does this formula imply that the Wornskian W(y1, y2) is either zero everywhere or nonzero everywhere?