Results 1 to 3 of 3

Math Help - Real Solution of 2nd Order ODE (Complex Roots)

  1. #1
    Member Maccaman's Avatar
    Joined
    Sep 2008
    Posts
    85

    Real Solution of 2nd Order ODE (Complex Roots)

    I've put the ODE into a system.

    We have:
    \left[\begin{array}{cc}\dot{y_1}\\\dot{y_2}\end{array}\r  ight] = \left[\begin{array}{cc}-1&-4\\1&3\end{array}\right] \left[\begin{array}{cc}y_1\\y_2\end{array}\right]

    which has Eigenvalues:
     \lambda_1 = -2 + i \sqrt{3}
     \lambda_2 = -2 - i \sqrt{3}

    and Eigenvectors:
    x^{(1)} = \left[\begin{array}{cc}1+i \sqrt{3}\\1\end{array}\right] , x^{(2)} = \left[\begin{array}{cc}1-i \sqrt{3}\\1\end{array}\right]

    General Solution is
     y(t) = c_1 \left[\begin{array}{cc}1+i \sqrt{3}\\1\end{array}\right] e^{(-2+i \sqrt{3})t} + c_2 \left[\begin{array}{cc}1-i \sqrt{3}\\1\end{array}\right]e^{(-2-i \sqrt{3})t}

    But this is where I am getting lost, I know that when the eigenvalues are complex I need to involve sin and cos, which is fine for a 1st order ODE but am getting a little stuck with this 2nd-order ODE.

    Also, I need to find the solution in Real Form. i.e.

     y(t) = \left[\begin{array}{cc}y_1(t)\\y_2(t)\end{array}\right]
    Thanks in advance to anyone who can help.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member Rebesques's Avatar
    Joined
    Jul 2005
    From
    At my house.
    Posts
    536
    Thanks
    10
    The real and imaginary parts of the general solution are two (linearly independent) real solutions.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member Maccaman's Avatar
    Joined
    Sep 2008
    Posts
    85
    I was wondering if someone could check my solution. If its not correct, I think it could be close.....

    General Solution:
    <br />
y(t) = c_1 \begin{pmatrix}1+i \sqrt{3}\\1\end{pmatrix} e^{(-2+i \sqrt{3})t} + c_2 \begin{pmatrix}1-i \sqrt{3}\\1\end{pmatrix}e^{(-2-i \sqrt{3})t}<br />

    <br />
= e^{-2t} \Bigg ( c_1 \begin{pmatrix}1+i \sqrt{3}\\1\end{pmatrix} e^{(\sqrt{3})it} + c_2 \begin{pmatrix}1-i \sqrt{3}\\1\end{pmatrix} e^{(-\sqrt{3})it} \Bigg )<br />

    <br />
= e^{-2t} \Bigg ( c_1 \begin{pmatrix}1+i \sqrt{3}\\1\end{pmatrix}(c_1 cos(\sqrt{3}t)+i sin(\sqrt{3}t)+c_2 \begin{pmatrix}1-i \sqrt{3}\\1\end{pmatrix}(cos(\sqrt{3}t)-i sin(\sqrt{3}t) \Bigg)<br />
(Complex Form)

    Note: Don't confuse  cos(\sqrt{3}t) and  sin(\sqrt{3}t) with  cos(\sqrt{3t}) and  sin(\sqrt{3t})

     = e^{-2t} \begin{pmatrix} a \\b+c \end{pmatrix}

    where  a = c_1 cos(\sqrt{3} t) + i c_1 sin (\sqrt{3} t) + c_2 cos(\sqrt{3} t) - i c_2 sin (\sqrt{3} t)

    and  b = (\sqrt{3})i \ c_1 cos(\sqrt{3} t) - c_1 cos(\sqrt{3} t) - (\sqrt{3}) c_1 sin(\sqrt{3} t) - i c_1 sin(\sqrt{3} t)

    and  c = - (\sqrt{3}) i c_2 cos(\sqrt{3} t) - c_2 sin(\sqrt{3} t) - (\sqrt{3}) c_2 sin(\sqrt{3} t) + i c_2 sin(\sqrt{3} t)

    (I'm sorry if this part is confusing, but when I tried to write one piece of latex code for this I got an error message saying something like 'Latex Error too big')

    = (\sqrt{3}+c_2)e^{-2t} \begin{pmatrix} cos(\sqrt{3} t) \\ -cos(\sqrt{3}t)-(\sqrt{3})sin(\sqrt{3}t) \end{pmatrix} ....

    .....  + i(c_1-c_2)e^{-2t} \begin{pmatrix} sin((\sqrt{3}t) \\ (\sqrt{3})cos(\sqrt{3}t)-sin(\sqrt{3}t) \end{pmatrix}<br />

     = c_1 e^{-2t} \begin{pmatrix}cos(\sqrt{3} t) \\ -cos(\sqrt{3}t)-(\sqrt{3})sin(\sqrt{3}t) \end{pmatrix}  + c_2 e^{-2t} \begin{pmatrix} sin((\sqrt{3}t) \\ (\sqrt{3})cos(\sqrt{3}t)-sin(\sqrt{3}t) \end{pmatrix}<br />
(Real Form)

    whew....(wipes brow)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Reapeated Real and Complex Roots
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: March 3rd 2011, 07:57 PM
  2. [SOLVED] Verify solution for 2nd order eqn. by taking real part....
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: April 5th 2010, 08:34 AM
  3. solution to the equation (complex roots)
    Posted in the Calculus Forum
    Replies: 4
    Last Post: January 16th 2010, 12:37 PM
  4. Replies: 5
    Last Post: May 6th 2009, 05:50 PM
  5. Higher Order Complex Roots
    Posted in the Calculus Forum
    Replies: 3
    Last Post: July 15th 2007, 06:38 PM

Search Tags


/mathhelpforum @mathhelpforum