# Lecture 15 - Homogeneous Linear Systems with Constant...

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Homogeneous Linear Systems with Constant Coefficients Solutions of Systems of ODEs

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Important Linear Algebra Recall Eigenvalues and Eigenvectors A v = λ v And Linear Independence and v 2 Are linearly independent if v 1 v 2 det = 0 v 1
Linear Systems of Ordinary Differential Equations a 22 a 12 a 11 a 21 y 1 y 1 y 2 y 2 + + y 0 1 y 0 2 = = Let’s rewrite this in matrix form: y 1 y 2 = a 22 a 12 a 11 a 21 y 0 1 y 0 2 Or y 0 = A y

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Linear Systems of Ordinary Differential Equations What if was an eigenvector of y A ? y 0 = A y Then: y 0 = A y
Linear Systems of Ordinary Differential Equations What if was an eigenvector of y A ? y 0 = A y Then: y 0 = y λ

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Linear Systems of Ordinary Differential Equations What if was an eigenvector of y A ? y 0 = A y Then: y 0 = y λ Or: y 1 y 2 = a 22 a 12 a 11 a 21 y 0 1 y 0 2
Linear Systems of Ordinary Differential Equations What if was an eigenvector of y A ? y 0 = A y Then: y 0 = y λ Or: y 1 y 2 = y 0 1 y 0 2 λ λ 0 0

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Linear Systems of Ordinary Differential Equations What if was an eigenvector of y A ?
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