Results 1 to 4 of 4

Math Help - System of first order differential equations

  1. #1
    Newbie
    Joined
    Apr 2010
    Posts
    9

    System of first order differential equations

    Hi
    i need help with this system of first order differential equations. Starting conditions are

    x_{1} (0) =1
    x_{2} (0) =1

    The system is:

     \frac{dx_{1}(t)}{dt}= -\frac{601}{400}x_{1}(t) + \frac{769}{1000}x_{2}(t)

     \frac{dx_{2}(t)}{dt}= -\frac{201}{400}x_{1}(t) -  \frac{1231}{1000}x_{2}(t)

    Thank you!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member Haven's Avatar
    Joined
    Jul 2009
    Posts
    197
    Thanks
    8
    Quote Originally Posted by magodiafano View Post
    Hi
    i need help with this system of first order differential equations. Starting conditions are

    x_{1} (0) =1
    x_{2} (0) =1

    The system is:

     \frac{dx_{1}(t)}{dt}= -\frac{601}{400}x_{1}(t) + \frac{769}{1000}x_{2}(t)

     \frac{dx_{2}(t)}{dt}= -\frac{201}{400}x_{1}(t) -  \frac{1231}{1000}x_{2}(t)

    Thank you!
    The system becomes
    \left[ \begin{matrix}\frac{dx_{1}(t)}{dt} \\ \frac{dx_{2}(t)}{dt}<br />
\end{matrix} \right] = \left[ \begin{matrix} -\frac{601}{400} & \frac{769}{1000} \\ -\frac{201}{400} &-  \frac{1231}{1000} \end{matrix} \right] \left[ \begin{matrix} x_1(t) \\ x_2(t) \end{matrix}\right]

    Now we pull out the matrix:
    A =\left[ \begin{matrix} -\frac{601}{400} & \frac{769}{1000} \\ -\frac{201}{400} &- \frac{1231}{1000} \end{matrix} \right]

    And now we assume the form of the solution:

    \left[ \begin{matrix} x_{1}(t) \\ x_{2}(t)<br />
\end{matrix} \right] = e^{\lambda_1t}\eta_1 + e^{\lambda_2t}\eta_2

    \lambda_1 and \lambda_2 are the eigenvalue of the matrix A.
    while \eta_1 and \eta_2 are the corresponding eigenvectors
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Apr 2010
    Posts
    9
    Quote Originally Posted by Haven View Post
    The system becomes
    \left[ \begin{matrix}\frac{dx_{1}(t)}{dt} \\ \frac{dx_{2}(t)}{dt}<br />
\end{matrix} \right] = \left[ \begin{matrix} -\frac{601}{400} & \frac{769}{1000} \\ -\frac{201}{400} &-  \frac{1231}{1000} \end{matrix} \right] \left[ \begin{matrix} x_1(t) \\ x_2(t) \end{matrix}\right]

    Now we pull out the matrix:
    A =\left[ \begin{matrix} -\frac{601}{400} & \frac{769}{1000} \\ -\frac{201}{400} &- \frac{1231}{1000} \end{matrix} \right]

    And now we assume the form of the solution:

    \left[ \begin{matrix} x_{1}(t) \\ x_{2}(t)<br />
\end{matrix} \right] = e^{\lambda_1t}\eta_1 + e^{\lambda_2t}\eta_2

    \lambda_1 and \lambda_2 are the eigenvalue of the matrix A.
    while \eta_1 and \eta_2 are the corresponding eigenvectors
    ehm i've several problem in computing autovectors... i used eig function in matlab but i'm not sure of the result! i'm not able to extract the eigenvector from the created matrix!
    what shall i do?
    I obtained this result

    eigenvalues = -1.3668 + 0.6067i
    -1.3668 - 0.6067i

    while eigenvectors:

    V =

    0.7777 0.7777
    0.1373 + 0.6135i 0.1373 - 0.6135i

    how could i use this result in order to find the solution of the system?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Apr 2010
    Posts
    9
    could someone help me?? it's very important!!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Solving a System of First Order Differential Equations
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: November 16th 2010, 10:08 PM
  2. System of first order differential equations
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: April 26th 2010, 05:11 AM
  3. System of first order differential equations
    Posted in the Calculus Forum
    Replies: 0
    Last Post: April 25th 2010, 08:29 PM
  4. Replies: 4
    Last Post: April 23rd 2010, 07:11 AM
  5. Replies: 2
    Last Post: November 25th 2008, 09:29 PM

Search Tags


/mathhelpforum @mathhelpforum