# Lecture 17 - Repeated Eigenvalues Linear Homogeneous...

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Repeated Eigenvalues Linear Homogeneous Systems of Equations

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Linear Homogeneous Systems of Equations If the matrix has eigenvectors and eigenvalues y 1 y 2 = a 22 a 12 a 11 a 21 y 0 1 y 0 2 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 If the eigenvalues are complex, use Euler’s Formula
Consider the System This matrix has the repeated eigenvalue y 1 y 2 = y 0 1 y 0 2 λ 1 v 1 and associated eigenvector 1 3 - 3 - 5 = - 2 = 1 1 So there is one specific solution e - 2 t = y 0 1 y 0 2 1 1

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Consider the System This matrix has the repeated eigenvalue and eigenvector y 1 y 2 = y 0 1 y 0 2 λ 1 v 1 1 3 - 3 - 5 = - 2 = 1 1 So there is one specific solution e - 2 t = y 0 1 y 0 2 1 1 But this isn’t enough to form a general solution.
Recall for second order linear equations, A r 2 + B r + C = 0 Has only one root, r 1 = - B 2 A General Solution takes the form: y = C 1 e r 1 t + C 2 te r 1 t A y 00 + B y 0 + C y = 0 If Characteristic Equation Inspiration from Repeated Roots

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Recall for second order linear equations, A r 2 + B r + C = 0 Has only one root, r 1 = - B 2 A General Solution takes the form: y = C 1 e r 1 t + C 2 te r 1 t A y 00 + B y 0 + C y = 0 If Characteristic Equation Inspiration from Repeated Roots
Perhaps a Second Solution Takes The Form = Let’s check!

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