AEM_3e_Chapter_10

AEM_3e_Chapter_10 - Part III Systems of Differential...

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Part III Systems of Differential Equations 10 10 Systems of Linear Differential Equations EXERCISES 10.1 Preliminary Theory 3. Let X = x y z . Then X 0 = 34 9 6 10 10 4 3 X . 6. Let X = x y z . Then X 0 = 34 0 590 016 X + e t sin2 t 4 e t cos2 t e t . 9. dx dt = x y +2 z + e t 3 t ; dy dt =3 x 4 y + z e t + t ; dz dt = 2 x +5 y +6 z e t t 12. Since X 0 = µ 5cos t 5sin t 2cos t 4sin t e t and µ 25 24 X = µ t t t t e t we see that X 0 = µ X . 15. Since X 0 = 0 0 0 and 121 6 1 2 1 X = 0 0 0 we see that X 0 = 6 1 2 1 X . 18. Yes, since W ( X 1 , X 2 )=8 e 2 t 6 = 0 the set X 1 , X 2 is linearly independent on −∞ <t< . 21. Since X 0 p = µ 2 1 and µ 14 32 X p + µ 2 4 t + µ 7 18 = µ 2 1 177
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10.1 Preliminary Theory we see that X 0 p = µ 14 32 X p + µ 2 4 t + µ 7 18 . 24. Since X 0 p = 3cos3 t 0 3sin3 t and 123 420 610 X p + 1 4 3 sin3 t = t 0 t we see that X 0 p = X p + 1 4 3 t. EXERCISES 10.2 Homogeneous Linear Systems 3. The system is X 0 = µ 42 5 2 2 X and det( A λ I )=( λ 1)( λ + 3) = 0. For λ 1 = 1 we obtain µ 52 0 5 2 1 0 = µ 0 00 0 so that K 1 = µ 2 5 . For λ 2 = 3 we obtain µ 12 0 5 2 5 0 = µ 0 0 so that K 2 = µ 2 1 . Then X = c 1 µ 2 5 e t + c 2 µ 2 1 e 3 t . 6. The system is X 0 = µ 62 31 X and det( A λ I )= λ ( λ + 5) = 0. For λ 1 = 0 we obtain µ 0 0 = µ 1 1 3 0 0 so that K 1 = µ 1 3 . λ 2 = 5 we obtain µ 0 36 0 = µ 1 2 0 0 so that K 2 = µ 2 1 . Then X = c 1 µ 1 3 + c 2 µ 2 1 e 5 t . 178
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10.2 Homogeneous Linear Systems 9. We have det( A λ I )= ( λ + 1)( λ 3)( λ + 2) = 0. For λ 1 = 1, λ 2 = 3, and λ 3 = 2 we obtain K 1 = 1 0 1 , K 2 = 1 4 3 , and K 3 = 1 1 3 , so that X = c 1 1 0 1 e t + c 2 1 4 3 e 3 t + c 3 1 1 3 e 2 t . 12. We have det( A λ I )=( λ 3)( λ + 5)(6 λ ) = 0. For λ 1 =3, λ 2 = 5, and λ 3 = 6 we obtain K 1 = 1 1 0 , K 2 = 1 1 0 , and K 3 = 2 2 11 , so that X = c 1 1 1 0 e 3 t + c 2 1 1 0 e 5 t + c 3 2 2 11 e 6 t . 21. We have det( A λ I λ 2) 2 =0.For λ 1 = 2 we obtain K = µ 1 1 .
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This note was uploaded on 01/03/2011 for the course BIS 511 at Yale.

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AEM_3e_Chapter_10 - Part III Systems of Differential...

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