solutions5bd

solutions5bd - as our eigenvector v = 1-2 i 1 = 1 1 + i -2...

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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 ² x . Solution. First we find the eigenvalues of the matrix A = ± - 4 5 - 1 - 2 ² by solving the characteristic equation: det ± λ + 4 - 5 1 λ + 2 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 ²± x 1 x 2 ² = ± 0 0 ² Both equations are multiples of the single equation x 1 +( - 1+2 i ) x 2 = 0. We can choose
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Unformatted text preview: as our eigenvector v = 1-2 i 1 = 1 1 + i -2 . Since complex eigenvalue-eigenvectors come in conjugate pairs, we have the eigenvalues-3 2 i with corresponding eigenvectors 1 1 i -2 We now have enough information to write down the general solution to the dierential equation: x ( t ) = c 1 e-3 t 1 1 cos 2 t--2 sin 2 t ! + c 2 e-3 t 1 1 sin 2 t + -2 cos 2 t 1...
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This note was uploaded on 03/05/2008 for the course MAE 33B taught by Professor Lee during the Spring '07 term at UCLA.

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