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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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 Spring '13
 EricCytrynbaum
 Differential Equations, Equations, Complex Numbers

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