Lecture 4 Notes

0 independence and the wronskian section 32

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online