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
    527
    Thanks
    7
    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