Suppose that the vectors x 1 (1) and x 2 (1) of Problem 34 are solutions of
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Suppose that the vectors x1 (1) and x2 (1) of Problem 34 are solutions of the equation x' = P(t)x, where the 2 x 2 matrix P(t) is continuous on the open interval I. Show that if there exists a point a of I at which their Wronskian W(a) is zero, then there exist numbers c and c not both zero such that c1x1 (a) + c2x2 (a) = 0. Then conclude from the uniqueness of solutions of the equation x' = P(t)x that
c1x1 (t) + c2x2 (t) = 0
for all t in I; that is, that x1 and x2 are linearly dependent. This proves part (b) of Theorem 2 in the case n = 2.
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Differential Equations And Linear Algebra
ISBN: 9780134497181
4th Edition
Authors: C. Edwards, David Penney, David Calvis
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