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MathSofiya Group Title

Solve using variation of parameters \[y''-2y'+y=e^{2x}\]

  • 2 years ago
  • 2 years ago

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  1. TuringTest Group Title
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    complimentary....

    • 2 years ago
  2. MathSofiya Group Title
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    meaning?

    • 2 years ago
  3. MathSofiya Group Title
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    \[y_c=c_1e^{-x}+c_2xe^{-x}\]

    • 2 years ago
  4. TuringTest Group Title
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    right...

    • 2 years ago
  5. TuringTest Group Title
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    now we need the wronskian, do you know about that?

    • 2 years ago
  6. MathSofiya Group Title
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    nope

    • 2 years ago
  7. TuringTest Group Title
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    it is the determinant of a matrix

    • 2 years ago
  8. TuringTest Group Title
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    for\[y_c=c_1y_1+c_2y_2\]the Wronskian is\[W(y_1,y_2)=\left|\begin{matrix}y_1&y_2\\y_1'&y_2'\end{matrix}\right|\]

    • 2 years ago
  9. TuringTest Group Title
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    that's a determinant in case you forgot, so find that determinant

    • 2 years ago
  10. MathSofiya Group Title
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    \[\left|\begin{matrix}e^{-x}&xe^{-x}\\-e^{-x}&-e^{-x}(x-1)\end{matrix}\right|\] =\[e^{-x}(e^{-x}(x-1))-xe^{-x}e^{-x}\]

    • 2 years ago
  11. MathSofiya Group Title
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    completely wrong?

    • 2 years ago
  12. MathSofiya Group Title
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    oh it's plus

    • 2 years ago
  13. TuringTest Group Title
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    yep you caught it :)

    • 2 years ago
  14. MathSofiya Group Title
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    \[e^{-x}(e^{-x}(x-1))+xe^{-x}e^{-x}\]

    • 2 years ago
  15. MathSofiya Group Title
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    that's it?

    • 2 years ago
  16. TuringTest Group Title
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    you dropped a minus

    • 2 years ago
  17. MathSofiya Group Title
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    where?

    • 2 years ago
  18. MathSofiya Group Title
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    ohhhh

    • 2 years ago
  19. MathSofiya Group Title
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    \[-e^{-x}(e^{-x}(x-1))+xe^{-x}e^{-x}\]

    • 2 years ago
  20. TuringTest Group Title
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    right

    • 2 years ago
  21. TuringTest Group Title
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    simplify and be happy :)

    • 2 years ago
  22. MathSofiya Group Title
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    does it equal\[ e^{-x}\]? \[-e^{-x}(e^{-x}(x-1))+xe^{-x}e^{-x}=e^{-x}\] \[{-x}(e^{-x}(x-1))+xe^{-x}=1\]

    • 2 years ago
  23. MathSofiya Group Title
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    \[(e^{-x}(x-1))+xe^{-x}=1\]

    • 2 years ago
  24. MathSofiya Group Title
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    typo

    • 2 years ago
  25. TuringTest Group Title
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    no, why did you set it equal to e^-x, we can't set it equal to anything yet...

    • 2 years ago
  26. MathSofiya Group Title
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    oh my bad haha

    • 2 years ago
  27. MathSofiya Group Title
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    so how should I simplify it then?

    • 2 years ago
  28. TuringTest Group Title
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    \[-e^{-x}(e^{-x}(x-1))+xe^{-x}e^{-x}=e^{-x}(e^{-x}(1-x))+xe^{-x}e^{-x}\]now try again while I try to eat ;)

    • 2 years ago
  29. MathSofiya Group Title
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    -(x-1)=1-x 1-x=1-x

    • 2 years ago
  30. MathSofiya Group Title
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    no no no ...wait a second

    • 2 years ago
  31. MathSofiya Group Title
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    now i get 2-x=2-x

    • 2 years ago
  32. MathSofiya Group Title
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    nope still wrong

    • 2 years ago
  33. MathSofiya Group Title
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    x=1

    • 2 years ago
  34. MathSofiya Group Title
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    stop laughing

    • 2 years ago
  35. MathSofiya Group Title
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    Let's try that again \[-e^{-x}(e^{-x}(x-1))+xe^{-x}e^{-x}=e^{-x}(e^{-x}(1-x))+xe^{-x}e^{-x}\]

    • 2 years ago
  36. MathSofiya Group Title
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    they're just equal to each other

    • 2 years ago
  37. MathSofiya Group Title
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    \[-e^{-x}(e^{-x}(x-1))+xe^{-x}e^{-x}\]

    • 2 years ago
  38. MathSofiya Group Title
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    \[-e^{-2x}(x-1)+xe^{-2x}\]

    • 2 years ago
  39. MathSofiya Group Title
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    \[e^{-2x}((1-x)+x)\]

    • 2 years ago
  40. MathSofiya Group Title
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    \[e^{-2x}\]

    • 2 years ago
  41. MathSofiya Group Title
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    yes?

    • 2 years ago
  42. TuringTest Group Title
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    yes, sorry, I was afk

    • 2 years ago
  43. MathSofiya Group Title
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    LOL ok

    • 2 years ago
  44. TuringTest Group Title
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    do you know the formula for the particular solution for variation of parameters?

    • 2 years ago
  45. MathSofiya Group Title
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    • 2 years ago
  46. MathSofiya Group Title
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    something like that?

    • 2 years ago
  47. TuringTest Group Title
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    almost unidentifiable to me in that form, but correct I'm sure I have the more direct formula memorized http://tutorial.math.lamar.edu/Classes/DE/VariationofParameters.aspx (the green box with the formula that says "variation of parameters)

    • 2 years ago
  48. MathSofiya Group Title
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    is the first one a y''? \[y''+q(t)y'+r(t)y=g(t)\]

    • 2 years ago
  49. TuringTest Group Title
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    yeah, it looks screwy I know

    • 2 years ago
  50. MathSofiya Group Title
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    do we know what q(t) and r(t) are? are they the coefficients of y' and y? Probably not it looks like

    • 2 years ago
  51. TuringTest Group Title
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    yes and in this case they are the coefficients, which in this case are constants the formula is just pointing out that they may not be

    • 2 years ago
  52. MathSofiya Group Title
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    oh I see, soo... \[y''+q(t)y'+r(t)y=g(t)\] \[y''-2y'+y=e^{2x}\] so now I integrate

    • 2 years ago
  53. TuringTest Group Title
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    right, using the formula... r(t) and r(t) don't even come into the formula for the particular solution in variation of parameters

    • 2 years ago
  54. TuringTest Group Title
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    q(t) and r(t)... they determine the complimentary only, which as you can see *is* part of the formula for the particular

    • 2 years ago
  55. MathSofiya Group Title
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    \[Y_p(t)=-e^{-x}\int\frac{xe^{-x}e^{-x}}{e^{-2x}(e^{-x},xe^{-x})}dx+xe^{-x}\int\frac{e^{-x}e^{2x}}{e^{-2x}(e^{-x},xe^{-x})}dx\]

    • 2 years ago
  56. TuringTest Group Title
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    the first integrand is wrong I think

    • 2 years ago
  57. TuringTest Group Title
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    I think you put the wrong g(x)

    • 2 years ago
  58. MathSofiya Group Title
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    \[Y_p(t)=-e^{-x}\int\frac{xe^{-x}e^{2x}}{e^{-2x}(e^{-x},xe^{-x})}dx+xe^{-x}\int\frac{e^{-x}e^{2x}}{e^{-2x}(e^{-x},xe^{-x})}dx\]

    • 2 years ago
  59. TuringTest Group Title
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    oh wait, wait, the denominator in each integrand is the wronskian, which we found to be e^(-2x) I think you are confusing notation with the parentheses

    • 2 years ago
  60. MathSofiya Group Title
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    Oh lol so the denominator is just e^{-2x}

    • 2 years ago
  61. TuringTest Group Title
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    right

    • 2 years ago
  62. TuringTest Group Title
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    notice that makes our integrals quite tolerable :)

    • 2 years ago
  63. MathSofiya Group Title
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    \[Y_p(t)=-e^{-x}\int\frac{xe^{-x}e^{2x}}{e^{-2x}}dx+xe^{-x}\int\frac{e^{-x}e^{2x}}{e^{-2x}}dx\] That's a lot easier to integrate

    • 2 years ago
  64. TuringTest Group Title
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    oh yeah :) especially after a little simplification

    • 2 years ago
  65. MathSofiya Group Title
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    \[-\frac{e^{2x}}{9}(3x-1)+\frac x3e^{3x}\]

    • 2 years ago
  66. MathSofiya Group Title
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    so what was the point of doing variation of parameters

    • 2 years ago
  67. TuringTest Group Title
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    well now you have a particular solution

    • 2 years ago
  68. TuringTest Group Title
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    y(x)=yc+yp you got you now so you're good to go

    • 2 years ago
  69. TuringTest Group Title
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    you got yp*

    • 2 years ago
  70. TuringTest Group Title
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    I did not check your integral btw

    • 2 years ago
  71. MathSofiya Group Title
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    that's ok I did wolfram

    • 2 years ago
  72. TuringTest Group Title
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    right-o, I'd do the same right now :)

    • 2 years ago
  73. MathSofiya Group Title
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    oh I see \[c_1e^{-x}+c_2xe^{-x}-\frac{e^{2x}}{9}(3x-1)+\frac x3e^{3x}\]

    • 2 years ago
  74. TuringTest Group Title
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    yep, and of course if this is an IVP we now proceed to find c1 and c2 otherwise we're done having found the general solution

    • 2 years ago
  75. MathSofiya Group Title
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    good, wow, that was a lot THanks Turing!

    • 2 years ago
  76. TuringTest Group Title
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    Always a pleasure Sofiya :D

    • 2 years ago
  77. Mendicant_Bias Group Title
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    Looking back over this, I don't understand how you guys got a negative double root to the characteristic equation. I got m_1 = m_2 = +1. Plugging into the normal formula for repeated roots, getting e^x and xe^x, not e^-x. Am I missing something? @ganeshie8

    • 6 days ago
  78. Mendicant_Bias Group Title
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    @jim_thompson5910 @agent0smith

    • 6 days ago
  79. Mendicant_Bias Group Title
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    @Zarkon @terenzreignz Is anybody else familiar with ODE's seeing this, or am I missing something?

    • 6 days ago
  80. Mendicant_Bias Group Title
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    @zepdrix Anybody?

    • 6 days ago
  81. zepdrix Group Title
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    @Mendicant_Bias For the homogeneous solution? Yes the characteristic equation appears to be giving +1 as a repeated root, as you indicated

    • 6 days ago
  82. zepdrix Group Title
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    So yah, that was a mistake early in the thread :C

    • 6 days ago
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