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RolyPoly

  • 3 years ago

Solve the initial problem \[y_1''=2y_1+y_2+y_1'+y_2'\]\[y_2''=-5y_1+2y_2+5y_1'-y_2'\]\(y_1(0)=y_2(0)=y_1'(0)=4\), \(y_2'(0)=-4\)

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  1. hba
    • 3 years ago
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    Integrate.

  2. RolyPoly
    • 3 years ago
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    Note that I put this under linear algebra.

  3. RolyPoly
    • 3 years ago
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    I think it has something to do with eigenvalues and eigenvectors

  4. RolyPoly
    • 3 years ago
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    and diagonalization too

  5. experimentX
    • 3 years ago
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    let y'1 = u, and y'2 = v, you get 4x4 system.

  6. RolyPoly
    • 3 years ago
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    Hmm.. How??

  7. experimentX
    • 3 years ago
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    |dw:1357743467559:dw|

  8. experimentX
    • 3 years ago
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    |dw:1357743488688:dw|

  9. experimentX
    • 3 years ago
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    let that matrix be A, you get X' = AX <-- this is logistic equation ... I must admit ... to me, this is not a nice question.

  10. experimentX
    • 3 years ago
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    the solution is \[ X = Se^{\Lambda t}S^{-1}X(0)\]

  11. experimentX
    • 3 years ago
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    |dw:1357745773804:dw|

  12. sirm3d
    • 3 years ago
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    how about laplace transform?

  13. experimentX
    • 3 years ago
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    |dw:1357746034677:dw||dw:1357746105903:dw|

  14. experimentX
    • 3 years ago
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    |dw:1357746162435:dw|

  15. experimentX
    • 3 years ago
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    |dw:1357746200518:dw||dw:1357746235140:dw|

  16. experimentX
    • 3 years ago
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    so the final solution is |dw:1357746290760:dw|

  17. vf321
    • 3 years ago
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    @sirm3d Using the laplace transform is not as pretty as @experimentX 's cleaner linear algebra solution. You're not guaranteed invertible functions in the frequency domain.

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