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# section%208.2 - Chapter 8 Systems of Linear First-Order...

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Chapter 8. Systems of Linear First-Order Di ff erential Equations § 8.2 Homogeneous Linear Systems 1. Homogeneous Linear Systems: X = AX where A is an n × n matrix of constants. 2. Eigenvalues and Eigenvectors. Def: The polynomial equation det( A λ I ) = 0 is called the characteristic equation of A . The solutions of this equation are called the eigenvalues of A ; A vector v satisfying the equation A v = λ v is called the eigenvector corresponding to λ . We only study X = AX where A is an 2 × 2 matrix of constants. Distinct Real Eigenvalues: Let λ 1 , λ 2 be two distinct real eigenval- ues of the coe cient matrix A of the homogeneous system X = AX and let v 1 , v 2 be the corresponding eigenvectors. Then the general so- lution of X = AX is X = c 1 v 1 e λ 1 t + c 2 v 2 e λ 2 t , where the c 1 and c 2 are arbitrary constants. Ex: Solve dx dt = 2 x + 3 y dy dt = 2 x + y 1

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Repeated Eigenvalues: Solve X = AX where A = 2 0 0 2 and A = 2 0 1 2 In the case that λ has two linearly independent eigenvectors v 1 and v 2 , the general solution is
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