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ChmE

  • 2 years ago

Method of undetermined coefficients...

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  1. ChmE
    • 2 years ago
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    Must be in the form\[ay''+by'+cy=Ct ^{m}e ^{rt}\]What does C stand for? I thought it was a constant but the equation\[y''+2y'-y=t ^{-1}e ^{t}\]Does not satisfy this method. i assumed C=1 so it would, bu tit doesn't. Explanation?

  2. ChmE
    • 2 years ago
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    @UnkleRhaukus

  3. ChmE
    • 2 years ago
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    Find a particular solution to the differential equation\[4y''+11y'-3y=-2te ^{-3t}\]

  4. across
    • 2 years ago
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    I believe that the method only applies for \(m\geq0\), but I cannot recall to save my life.

  5. ChmE
    • 2 years ago
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    \[4r ^{2}+11r-3=0\rightarrow (4r-1)(r+3)\rightarrow r= \frac{ 1 }{ 4 },-3\]

  6. ChmE
    • 2 years ago
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    That makes sense I'm going to look that up. Thanks

  7. ChmE
    • 2 years ago
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    @across I found it you are right

  8. ChmE
    • 2 years ago
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    since -3 does appear as a root s=1 which is the multiplicity of the root -3

  9. ChmE
    • 2 years ago
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    Ok @UnkleRhaukus I could use a little help on my last problem of the night. I posted the question in this stream 3rd from the top.

  10. ChmE
    • 2 years ago
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    \[Y _{p}= t(Ae ^{-3t})\]Do I have to include the other root as well?

  11. ChmE
    • 2 years ago
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    @Directrix @Hero Can I have a little push in the right direction

  12. ChmE
    • 2 years ago
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    @Hero do you see a mistake I have made or is anything coming to you about what to do next?

  13. Hero
    • 2 years ago
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    Sorry, I'm busy doing my own work at the moment.

  14. ChmE
    • 2 years ago
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    The answer in the back of the book is \[(\frac{ t }{ 13 }-\frac{ 8 }{ 169 })te ^{-3t}\]

  15. ChmE
    • 2 years ago
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    Oh, I understand

  16. ChmE
    • 2 years ago
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    @satellite73 @.Sam.

  17. ChmE
    • 2 years ago
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    @UnkleRhaukus Just in time I was about to give up. Do you mind helping me with this last question?

  18. UnkleRhaukus
    • 2 years ago
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    where are you up to

  19. ChmE
    • 2 years ago
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    I feel like I need to plug in \[Y _{p},Y'_{p}, Y''_{p}\]into the original to solve for A. I am up to writing an equation for Yp

  20. ChmE
    • 2 years ago
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    I feel like I have my roots and s correct

  21. UnkleRhaukus
    • 2 years ago
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    \[4y''+11y'-3y=-2te ^{-3t}\] \[4r^2+11r-3=0\]\[(4r-1)(r+3)=0\]\[r=1/4,-3\] \[y_c=ae^{t/4}+be^{-3t}\]

  22. ChmE
    • 2 years ago
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    Oh ya I forgot, when you have two diff roots you need to use two different unknowns. What about the t^s? My teacher threatened us if we forgot that on the test

  23. ChmE
    • 2 years ago
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    As i understand it t^s, where s is the multiplicity of your roots needs to be out in front.

  24. UnkleRhaukus
    • 2 years ago
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    \[y_p=t\left(c\cdot e^{-3t}+d\cdot t\cdot e^{-3t}\right)\]

  25. ChmE
    • 2 years ago
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    Why do you have two terms in the ( ) if m=1 shouldn't there be 1. And I don't mean to come across like I'm questioning you, just trying to undersstand

  26. UnkleRhaukus
    • 2 years ago
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    its a second order equation there will be two solutions

  27. ChmE
    • 2 years ago
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    So I see your Yp matches the books except I need to now solve for c and d by plugging Yp into the original equation right?.........Ok

  28. UnkleRhaukus
    • 2 years ago
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    take the first and second derivative of y_p

  29. ChmE
    • 2 years ago
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    Right then plug it in. I can do that on my own later. Thanks for your help again. I'm calling it a night

  30. UnkleRhaukus
    • 2 years ago
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    see you next time

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