Verify that y 1 = x and y 2 = x 2 are linearly independent solutions on
Question:
Verify that y1 = x and y2 = x2 are linearly independent solutions on the entire real line of the equation
x2y'' - 2xy' + 2y = 0,
but that W(x , x2) vanishes at x = 0. Why do these observations not contradict part (b) of Theorem 3?
Transcribed Image Text:
THEOREM 3 Wronskians of Solutions Suppose that y₁, y2, ..., yn are n solutions of the homogeneous nth-order linear equation y() + P₁(x) y(n-1)+...+ Pn-1(x)y' + Pn(x) y = 0 on an open interval I, where each p, is continuous. Let (a) If y₁, 2, (b) If y₁. Y2.. **** (3) W = W(y₁, y2...., yn). yn are linearly dependent, then W = 0 on I. yn are linearly independent, then W #0 at each point of I. Thus there are just two possibilities: Either W = 0 everywhere on I, or W #0 everywhere on I.
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Related Book For
Differential Equations And Linear Algebra
ISBN: 9780134497181
4th Edition
Authors: C. Edwards, David Penney, David Calvis
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