Lecture 16 - Complex Eigenvalues and Phase Portraits...

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Complex Eigenvalues and Phase Portraits
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Fundamental Set of Solutions For Linear System of ODEs y 1 y 2 = a 22 a 12 a 11 a 21 y 0 1 y 0 2 With Eigenvalues and Eigenvectors λ 1 v 1 and and v 2 λ 2 The General Solution Takes The Form y 1 y 2 = C 1 C 2 + e λ 1 t v 1 e λ 2 t v 2
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Not All Matrices Have Real Eigenvalues/Eigenvectors y 1 y 2 = y 0 1 y 0 2 - 1 1 - 4 - 1 Has Eigenvalues and Eigenvectors λ 1 v 1 and = - 1 + 2 i = 2 i 1 and = = , 1 - 2 i λ 2 v 2 - 1 - 2 i Has The General Solution C 1 e ( - 1+2 i ) t C 2 e ( - 1 - 2 i ) t + 2 i 1 1 - 2 i y =
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Not All Matrices Have Real Eigenvalues/Eigenvectors y 1 y 2 = y 0 1 y 0 2 - 1 1 - 4 - 1 Has The General Solution C 1 e ( - 1+2 i ) t C 2 e ( - 1 - 2 i ) t + 2 i 1 1 - 2 i y = Lots of Complex Numbers We want a Real General Solution
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Not All Matrices Have Real Eigenvalues/Eigenvectors y 1 y 2 = y 0 1 y 0 2 - 1 1 - 4 - 1 Has The General Solution C 1 e ( - 1+2 i ) t C 2 e ( - 1 - 2 i ) t + 2 i 1 1 - 2 i y = Lots of Complex Numbers We want a Real General Solution Recall Euler’s Formula
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Not All Matrices Have Real Eigenvalues/Eigenvectors y 1 y 2 = y 0 1 y 0 2 - 1 1 - 4 - 1 Has The General Solution C 1 e ( - 1+2 i ) t C 2 e ( - 1 - 2 i ) t + 2 i 1 1 - 2 i y = y = - 2 sin (2 t ) cos (2 t ) e - 1 t C 1 C 2 + ( ) e - 1 t C 1 C 2 ( ) + i - 2 cos (2 t ) sin (2 t ) Or Just a Constant
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Not All Matrices Have Real Eigenvalues/Eigenvectors y 1 y 2 = y 0 1 y 0 2 - 1 1 - 4 - 1 Has The General Solution C 1 e ( - 1+2 i ) t C 2 e ( - 1 - 2 i ) t + 2 i 1
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