solutions5bd - Math 33b Quiz 5bd Name UCLA ID 1 Find the...

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Math 33b, Quiz 5bd, May 24, 2007 Name: UCLA ID: 1. Find the general solution to the system of differential equations d x dt = - 4 5 - 1 - 2 x . Solution. First we find the eigenvalues of the matrix A = - 4 5 - 1 - 2 by solving the characteristic equation: det λ + 4 - 5 1 λ + 2 = 0 ( λ + 4)( λ + 2) - (1)( - 5) = 0 λ 2 + 6 λ + 13 = 0 λ = - 6 ± 36 - 52 2 = - 3 ± 2 i We have a pair of complex conjugate eigenvalues λ = - 3 ± 2 i. We search for the eigen- vector corresponding to λ = - 3 + 2 i : (( - 3 + 2 i ) I - A ) v = 0 1 + 2 i - 5 1 - 1 + 2 i x 1 x 2 = 0 0 Both equations are multiples of the single equation
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Unformatted text preview: as our eigenvector v = ± 1-2 i 1 ² = ± 1 1 ² + i ±-2 ² . Since complex eigenvalue-eigenvectors come in conjugate pairs, we have the eigenvalues-3 ± 2 i with corresponding eigenvectors ± 1 1 ² ± i ±-2 ² We now have enough information to write down the general solution to the differential equation: x ( t ) = c 1 e-3 t ³ ± 1 1 ² cos 2 t-±-2 ² sin 2 t ! + c 2 e-3 t ´ ± 1 1 ² sin 2 t + ±-2 ² cos 2 t µ 1...
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  • Spring '07
  • lee
  • general solution, complex conjugate eigenvalues

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