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Need help with 4th order differential equation

  1. Jul 19, 2012 #1
    1. The problem statement, all variables and given/known data

    y^(4)+y=0

    2. Relevant equations

    Need to use de Moivre's formula to obtain answer.

    3. The attempt at a solution.

    I can obtain the characteristic equation r^4=-1. From there I tried saying y_1 = cosx and y_2=sinx. However, my book has the answer listed as y= [e^(√(2)x/2)(cos(√(2)/2)x+sin(√(2)/2)x)]+[e^-(√(2)x/2)(cos(√(2)/2)x+sin(√(2)/2)x)]. I am unsure how they obtained this answer. When I asked my teacher he said I had to use the de Moivre formula, but I am unsure how to apply this. PLEASE HELP!
     
  2. jcsd
  3. Jul 20, 2012 #2

    vela

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    How did you go from r4=-1 to sines and cosines? What are the roots of the characteristic equation?
     
  4. Jul 20, 2012 #3
    I said r^2(r^2)=0 so I took the first root to be (0+i) and the second to be (0-i), from there I can say y=e^(0)(Ccosx+C'sinx). I was unsure how to get the other two roots from there. Tried reduction of order saying that if y_1=cosx then v_1y_1=y_3 but this does not give the answer in the book.
     
  5. Jul 20, 2012 #4

    vela

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    You need to solve for r. You've only gotten to ##r^2 = \pm i## so far.
     
  6. Jul 20, 2012 #5
    I know, I only knew how to solve for the two of them. I tried to find the other two using the reduction of order method and was unable to get the answer the book said.
     
  7. Jul 20, 2012 #6

    vela

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    You haven't solved for any of them yet.
     
    Last edited: Jul 20, 2012
  8. Jul 20, 2012 #7
    That's what you need to do then. You have the equation, and I'll use z:

    [tex]z^4=-1[/tex]

    or:

    [tex]z=\sqrt[4]{-1}[/tex]

    and de Moivre's formula allows you to solve for the four roots using the expression:

    [tex]z=|r|^{1/4}e^{i/4(\theta+2n\pi)}[/tex]

    but I think though you may not know how to apply it. r is the absolute value of -1 or just 1 right. Theta is the argument of -1. Well, that's just pi. And n goes from 0 to 3.
     
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