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Math Help - eigenvector, complex roots

  1. #1
    Newbie
    Joined
    Oct 2009
    Posts
    24

    eigenvector, complex roots

    I am asked to solve the system:

    dx/dt =
    -2 -2
    4 2

    with the initial condition
    x(0) =
    -1
    -1

    I find the roots of the characteristic equation to be +/- 2i
    and the associated eigenvector to be
    1
    2-2i

    from euler's formula, i find that x(t) = [cos(2t)+isin(2t)](1 : 2-2i)
    x(t) =
    cos2t + isin2t
    2cos2t-2icos2t+2isin2t+2sin2t

    x(t) =
    cos2t
    2cos2t + 2sin2t

    +
    (i)
    sin2t
    -2cos2t+2sin2t

    with the initial condition, i find c1 = -3 and c2 = -5/2.

    Where am I wrong (submitted answer is not correct)
    Thanks, let me know if anything needs more explanatioin.
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  2. #2
    Junior Member
    Joined
    Apr 2009
    Posts
    73
    Quote Originally Posted by plopony View Post
    I am asked to solve the system:

    dx/dt =
    -2 -2
    4 2

    with the initial condition
    x(0) =
    -1
    -1

    I find the roots of the characteristic equation to be +/- 2i
    and the associated eigenvector to be
    1
    2-2i

    from euler's formula, i find that x(t) = [cos(2t)+isin(2t)](1 : 2-2i)
    x(t) =
    cos2t + isin2t
    2cos2t-2icos2t+2isin2t+2sin2t

    x(t) =
    cos2t
    2cos2t + 2sin2t

    +
    (i)
    sin2t
    -2cos2t+2sin2t

    with the initial condition, i find c1 = -3 and c2 = -5/2.

    Where am I wrong (submitted answer is not correct)
    Thanks, let me know if anything needs more explanatioin.
    Firstly, the associated eigenvector is not
    1
    2i
    It is
    1
    -1-i.

    Now, letting your eigenvalue be \omega=\alpha +\beta i, and your eigenvector to be w=z_1+z_2i , you have that
    <br />
X(t)= c_1e^{\alpha t}(cos(\beta t)z_1-sin(\beta t)z_2) +c_2e^{\alpha t}(cos(\beta t)z_2+sin(\alpha t) z_1)

    Here Your \alpha is zero, so he exponentials become one. Your z_1 = 1,-1;z_2=0,-1
    Hence, substituting your values in you get
    x(t) = c_1 cos(2t)+c_2 sin(2t); y(t) =-c_1 cos(2t) +c_1sin(2t) -c_2cos(2t)-c_2sin(2t)
    Using x(0) =-1, y(0) =-1
    c_1=-1, c_2 =2

    Can you work through all the steps? I would write it in matrix form, but am unable to using latex, I don't know how. And am too busy to hunt down how. So I called the top line of the Matrix X(t) x(t) and the bottom line y(t). I have also skipped a fair amount of the calculations, but hopefully you will be able to work your way through yourself.
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  3. #3
    Newbie
    Joined
    Oct 2009
    Posts
    24
    thanks for the reply, I appreciate it. I think since you point out i got the eigenvector wrong, i should be all set.
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