Homework 12 - M341 H12(S Zhang 5.2-3 1 Solve the system by...

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M341 H12 (S. Zhang) 5.2-3. 1. Solve the system by eigenfunctions: x 1 = 2 x 1 + 3 x 2 , x 2 = 2 x 1 + x 2 ans: (5.2:2) det( A - λI ) = 0 , λ = - 1 , 4 For λ = - 1 3 3 | 0 2 2 | 0 x = C 1 - 1 For λ = 4 - 2 3 | 0 2 - 3 | 0 x = C 3 2 x = x 1 x 2 = C 1 e - t 1 - 1 + C 2 e 4 t 3 2 2. Solve the system by eigenfunctions: x 1 = 9 x 1 + 5 x 2 , x 2 = - 6 x 1 + 2 x 2 x 1 (0) = 1 , x 2 (0) = 0 ans: (5.2:6) det( A - λI ) = 0 , λ = 3 , 4 For λ = 3 6 5 | 0 - 6 - 5 | 0 x = C 5 - 6 For λ = 4 5 5 | 0 - 6 - 6 | 0 x = C 1 - 1 x = x 1 x 2 = C 1 e 3 t 5 - 6 + C 2 e 4 t 1 - 1 By the initial conditions, C 1 = - 1 , C 2 = 6 So x 1 = - 5 e 3 t + 6 e 4 t , x 2 = 6 e 3 t - 6 e 4 t 3. Solve the system by eigenfunctions: x 1 = x 1 - 5 x 2 , x 2 = x 1 - x 2 ans: (5.2:8) det( A - λI ) = 0 , λ = ± 2 i For λ = 2 i 1 - 2 i - 5 | 0 1 - 1 - 2 i | 0 x = C 5 1 - 2 i x = x 1 x 2 = C 1 cos 2 t 5 1 - sin 2 t 0 - 2 + C 2 cos 2 t 0 - 2 + sin 2 t 5 1 4. Solve the system by eigenfunctions: x 1 = 5 x 1 - 9 x 2 , x 2 = 2 x 1 - x 2 ans: (5.2:13) det( A - λI ) = 0 , λ = 2 ± 3 i For λ = 2 + 3 i 3 - 3 i - 9 | 0 2 - 3 - 3 i | 0 x = C 9 3 - 3 i x = x 1 x 2 = C 1 e 2 t cos 3 t 9 3 - sin 3 t 0 - 3 + C 2 e 2 t cos 3 t 0 - 3 + sin 3 t 9 3 5. Solve the system x = Ax by eigenfunctions: A = 1 2 2 0 3 3 0 0 4
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