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**Unformatted text preview: **Chapter 8. Systems of Linear First-Order Diﬀerential 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 − λI ) = 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 Av = λv 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 λ1 , λ2 be two distinct real eigenvalues of the coeﬃcient matrix A of the homogeneous system X = AX and let v1 , v2 be the corresponding eigenvectors. Then the general solution of X = AX is X = c1 v1 eλ1 t + c2 v2 eλ2 t , where the c1 and c2 are arbitrary constants. Ex: Solve
dx dt dy dt = 2x + 3y = 2x + y 1 Repeated Eigenvalues: Solve X = AX where A = 20 02 and A = 20 12 In the case that λ has two linearly independent eigenvectors v1 and v2 , the general solution is X = c1 v1 eλt + c2 v2 eλt , where the c1 and c2 are arbitrary constants. In the case that λ has only one linearly independent eigenvector v , ﬁnd the second vector w by solving the following equation (λI − A)w = v and the general solution is X = c1 veλt + c2 (vteλt + weλt ), where the c1 and c2 are arbitrary constants. Complex Roots: λ = α ± β i. Let v= x0 + x1 i y0 + y1 be the eigenvector corresponding to α + β i. The the general solution is X = c1 X1 + c2 X2 where X1 = x0 cos β t − x1 sin β t αt e, y0 cos β t − y1 sin β t X2 = 28 −1 −2 x0 sin β t + x1 cos β t αt e y0 sin β t + y1 cos β t Ex: Solve X = AX where A = 2 ...

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