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.

Transcribed Image Text:

THEOREM 2 Wronskians of Solutions Suppose that X₁, X₂, ..., X, are n solutions of the homogeneous linear equation x = P(1)x on an open interval I. Suppose also that P(t) is continuous on I. Let W = W(X1, X2,..., Xn). Then • If X1, X2, ..., Xn are linearly dependent on I, then W = 0 at every point of I. If X1, X2, ..., X, are linearly independent on I, then W #0 at each point of I. Thus there are only two possibilities for solutions of homogeneous systems: Ei- ther W = 0 at every point of I, or W = 0 at no point of I.