Lecture 4 Notes

# 0 independence and the wronskian section 32

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Unformatted text preview: han t0=0, we require that ￿ y1 (t0 )y2 (t0 ) − ￿ y1 (t0 )y2 (t0 ) ￿= 0 Independence and the Wronskian (Section 3.2) • For any two solutions to some linear ODE, to ensure that we have a general solution, we need to check that ￿ ￿ y1 (0) y2 (0) ￿ ￿ det ￿ = y1 (0)y2 (0) − y1 (0)y2 (0) ￿= 0 ￿ y1 (0) y2 (0) • For ICs other than t0=0, we require that ￿ y1 (t0 )y2 (t0 ) − ￿ y1 (t0 )y2 (t0 ) • This quantity is called the Wronskian. ￿= 0 Independence and the Wronskian (Section 3.2) 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. 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. Find values of C1≠0 and C2≠0 so that C1y1(t) + C2y2(t) = 0. (A) C1 = e−2t−3 , C2 = −e−2t+3 (B) C1 = e (C) C1 = e−3 , C2 = e3 (D) C1 = e−3 , C2 = −e3 (E) C1 = e3 , C2 = −e−3 −2t+3 , C2 = −e −2t−3 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. Find values of C1≠0 and C2≠0 so that C1y1(t) + C2y2(t) = 0. (A) C1 = e−2t−3 , C2 = −e−2t+3 (B) C1 = e (C) C1 = e−3 , C2 = e3 (D) C1 = e−3 , C2 = −e3 (E) C1 = e3 , C2 = −e−3 −2t+3 , C2 = −e −2t−3 Independence and the Wronskian (Section 3.2) 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 th...
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## This note was uploaded on 02/12/2014 for the course MATH 256 taught by Professor Ericcytrynbaum during the Spring '13 term at UBC.

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