HW7.EGM4313.Solutions - Week 7 Problems 1 Find the general...

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Week 7 Problems: 1) Find the general solution in real form to x x - - - - = 0 1 2 0 1 1 2 0 3 . Solution: First find the eigenvalues: 2 1 , 2 0 6 7 4 1 2 0 1 1 2 0 3 2 3 i ± - - = = - - - - = - - - - - - - λ λ λ λ λ λ λ Next find the eigenvectors: - = - = = = - - - - = 1 2 2 , 2 0 0 0 2 1 2 0 1 1 2 0 1 : 2 2 1 3 1 3 2 1 x x x x x x x x λ is an eigenvector. ( 29 + - = + - = = + = = - - - - - - + - = 2 0 2 1 1 0 2 1 1 2 2 , 2 2 1 0 0 0 2 1 1 2 0 2 1 2 0 2 2 : 2 1 2 1 1 3 3 2 1 i i i x x i x x i x x x x i i i i λ The general solution is: ( 29 ( 29 ( 29 ( 29 t t t e t t C e t t C e C x - - - + - + - - + - = 2 cos 2 0 2 2 sin 1 1 0 2 sin 2 0 2 2 cos 1 1 0 1 2 2 3 2 2 1 2) Find the solution to the initial value problem
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( 29 - - = - = 30 2 1 0 , 2 6 3 0 1 4 0 0 1 x x x Solution: Eigenvalues are λ =1, 1, 2 by inspection.
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