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Thread: Roots of this auxiliary eqn?

  1. #1
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    Roots of this auxiliary eqn?

    I'm trying to find the auxiliary roots of this DE

    $\displaystyle y'''+iy=0$
    So the auxiliary eqn is
    $\displaystyle m^3+i=0$
    and m=i ?
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  2. #2
    Super Member Random Variable's Avatar
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    Use De Moivre's formula to find the three roots.

    EDIT: $\displaystyle -i = e^{-i \pi /2} = \cos(-\pi/2) + i \sin(-\pi/2) $

    $\displaystyle (-i)^{1/3} = e^{i(-\pi/6 + 2k \pi/3)} = \cos (-\pi/6 + 2k \pi /3) + i \sin(-\pi/6 + 2k \pi /3)$ for k = 0, 1, and 2

    When k=0, $\displaystyle (-i)^{1/3} = \cos(-\pi/6) + i \sin(-\pi/6) = \frac{\sqrt{3}}{2} - \ \frac{i}{2} $

    when k =1, $\displaystyle (-1)^{1/3} = \cos(\pi/2) + i \sin(\pi/2) = i $

    and for the k=2, $\displaystyle (-1)^{1/3} = -\frac{\sqrt{3}}{2} - \ \frac{i}{2} \ $

    The complex roots are not conjugate pairs like in the other problem, so the general solution is $\displaystyle y(x) = C_{1}e^{(\frac{\sqrt{3}}{2} - \ \frac{i}{2})x} + C_{2}e^{ix} + C_{3}e^{(-\frac{\sqrt{3}}{2} - \ \frac{i}{2} )x}$


    BTW, Wolfram Alpha (the web site associated with Mathematica) can solve differential equations like these.
    Last edited by Random Variable; Mar 21st 2010 at 01:36 PM.
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  3. #3
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    Ah that formula is new to me. Just to be clear the $\displaystyle -i = e^{-\pi /2} $ Comes from Euler's formula and $\displaystyle \cos(-\pi/2) + i \sin(-\pi/2)$ comes from DeMoivre...is it $\displaystyle \frac{\pi}{2}$ because we're dealing with just $\displaystyle i$?
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  4. #4
    Super Member Random Variable's Avatar
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    Quote Originally Posted by mmattson07 View Post
    Ah that formula is new to me. Just to be clear the $\displaystyle -i = e^{-\pi /2} $ Comes from Euler's formula and $\displaystyle \cos(-\pi/2) + i \sin(-\pi/2)$ comes from DeMoivre...is it $\displaystyle \frac{\pi}{2}$ because we're dealing with just $\displaystyle i$?

    It should be $\displaystyle -i=e^{-i\pi/2} = \cos (-\pi/2) + i \sin (-\pi /2) $

    Then $\displaystyle (-i)^{1/3} = e^{i (-\pi/6 + 2k \pi/3)} $

    But the rest is correct.


    I used De Moivre's formula to find the roots for the other problem, too.

    $\displaystyle -1= e^{i \pi} = \cos \pi + i \sin \pi $

    so $\displaystyle (-4)^{-1/4} = 4^{1/4}(-1)^{1/4} = \sqrt{2}e^{i(\pi /4 + 2k \pi /4)} = \sqrt{2} [(\cos(\pi /4 + 2k \pi /4) + i \sin (\pi /4 + 2k \pi /4)]$

    Then let k =0,1,2,and 3 to find the four roots.


    http://en.wikipedia.org/wiki/De_Moivre's_formula
    Last edited by Random Variable; Mar 21st 2010 at 01:40 PM.
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  5. #5
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    Ok I see thanks for the link. Why is x $\displaystyle =\frac{\pi}{2}$ in this case?
    $\displaystyle -i=e^{-i\pi/2} = \cos (-\pi/2) + i \sin (-\pi /2)$
    Last edited by mmattson07; Mar 21st 2010 at 08:42 PM.
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  6. #6
    Super Member Random Variable's Avatar
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    Quote Originally Posted by mmattson07 View Post
    Ok I see thanks for the link. Why is x $\displaystyle =\frac{\pi}{2}$ in this case?
    $\displaystyle -i=e^{-i\pi/2} = \cos (-\pi/2) + i \sin (-\pi /2)$
    It's because of the location of -i on the complex plane.

    I'll try to explain, but you should really study up a bit on complex numbers.


    All real numbers (a+0i) are located along the x axis, all purely imaginary numbers (0+bi) are located along the y axis, and all complex numbers (a+bi) are located everywhere else.

    $\displaystyle -i = 0-i$ is located on the negative y axis and can be represented as a vector of unit length (since $\displaystyle \sqrt{0^{2}+1^{2}} = 1$) with it's tail at the origin and it's head at y=-1 .

    The angle (going counterclockwise) that the vector makes with the positive horizontal axis is $\displaystyle 3 \pi/2 $. But you usually want the angle to be between $\displaystyle \pi $and $\displaystyle -\pi $. The angle in that range that is equal to $\displaystyle 3 \pi /2 $ is $\displaystyle =-\pi/2 $

    You can write any number (real, imaginary, or complex) in the form $\displaystyle re^{i\theta} = r(cos \theta + i \sin \theta) $ where $\displaystyle r$ is the distance of the point from the origin in the complex plane and $\displaystyle \theta$ is the angle formed by the vector and the positive x axis.

    So $\displaystyle -i= 1e^{-i \pi /2} = 1(\cos -\pi/2 + i \sin -\pi/2) = 1(0-i) = -i$

    For the other problem the angle is $\displaystyle \pi $ because -1 is located on the negative x axis.
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  7. #7
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    Should it be $\displaystyle i$ or $\displaystyle -i$? Because $\displaystyle (i)^3=-i$ satisfying $\displaystyle m^3+i=0$ and $\displaystyle (-i)^3=i$ which would give $\displaystyle 2i=0$
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  8. #8
    Super Member Random Variable's Avatar
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    Quote Originally Posted by mmattson07 View Post
    Should it be $\displaystyle i$ or $\displaystyle -i$? Because $\displaystyle (i)^3=-i$ satisfying $\displaystyle m^3+i=0$ and $\displaystyle (-i)^3=i$ which would give $\displaystyle 2i=0$
    You're correct. $\displaystyle -i $ is not a root. But I never said it was. We're looking for the three roots of $\displaystyle (-i)^{1/3} $. I need to first express $\displaystyle -i$ in complex polar form (which I did above) before I can apply De Moivre's formula to find the roots. One of those roots is $\displaystyle i$.
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  9. #9
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    OH I see you get $\displaystyle (-i)^{1/3}$ from solving for m then apply DeMoivre. Ok thanks.
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