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ragman

  • 2 years ago

needed help with solving linear time invariant differential equations

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  1. ragman
    • 2 years ago
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    obtain the solution x(t) of the differential equation

  2. ragman
    • 2 years ago
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    |dw:1347265100080:dw|

  3. UnkleRhaukus
    • 2 years ago
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    \[\ddot x+\omega_n^2x=t\]

  4. ragman
    • 2 years ago
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    yup thats what i tried to write

  5. UnkleRhaukus
    • 2 years ago
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    \[m^2+\omega_n^2=0\]

  6. ragman
    • 2 years ago
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    would it be possible to explain that ?

  7. UnkleRhaukus
    • 2 years ago
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    the characteristic equation ?

  8. ragman
    • 2 years ago
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    so the first thing you would do is use the laplace transform on everything right im just not sure how to figure out how to do it for the w shaped variable

  9. UnkleRhaukus
    • 2 years ago
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    omega ? \(\omega_n^2\) is a constant for constant n

  10. UnkleRhaukus
    • 2 years ago
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    im not sure how you are going to use Laplace if you dont have any initial values

  11. ragman
    • 2 years ago
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    |dw:1347266422040:dw|

  12. ragman
    • 2 years ago
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    i knew i was forgetting something

  13. ragman
    • 2 years ago
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    and yeah i meant omega

  14. UnkleRhaukus
    • 2 years ago
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    well if we are going to use laplace transform you can forget the characteristic equation

  15. ragman
    • 2 years ago
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    ok sounds good the only difficulty i face is when things like omega come up i just get confused as to how im supposed to find the laplace transform of it

  16. UnkleRhaukus
    • 2 years ago
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    its just a positive constant, if you want to set \[\omega_n^2=A,\qquad A>0\]

  17. ragman
    • 2 years ago
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    ok im sorta getting it so what would the laplace transform of this constant be

  18. UnkleRhaukus
    • 2 years ago
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    the laplace transform of a constant times a function is the constant times the transfer \[\mathcal L\{αf(t)\} = α\mathcal L\{f(t)\}\] the laplace transform of one is one on the the transfer parameter \[\mathcal L\{1\}(s)=\frac 1s\]

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spraguer (Moderator)
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