Results 1 to 2 of 2

Math Help - Solve DE using eigenvalues

  1. #1
    MHF Contributor alexmahone's Avatar
    Joined
    Oct 2008
    Posts
    1,074
    Thanks
    7

    Solve DE using eigenvalues

    Solve the following set of differential equations (using eigenvalues and eigenvectors) where u is a vector of dimension 2.

    \displaystyle \frac{du}{dt}=\left[ \begin{array}{cc} 1 & 0 \\ 1 & 1 \end{array} \right]u

    My attempt: The eigenvalues of A are 1, 1. \displaystyle \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] is the only eigenvector. How do I proceed?
    Last edited by alexmahone; March 19th 2011 at 12:02 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    Let

    A=\begin{bmatrix}1 &0\\1 &1\end{bmatrix}.

    Your solution is u=e^{tA}u_{0}.

    So you must compute e^{tA}. But how? It turns out that with this particular A, you can easily compute the nth power, thus enabling you to compute

    e^{tA}=\displaystyle\sum_{n=0}^{\infty}\frac{(tA)^  {n}}{n!}.

    Normally, you'd use the eigenvalue expansion. But, as you've noticed, you can't do that with this matrix. You could try to use generalized eigenvectors. Another approach is to try to compute the nth power directly, which I think you can do here. What do you get when you compute the nth power of A?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Eigenvalues
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: April 27th 2010, 01:08 PM
  2. eigenvalues
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: April 26th 2010, 02:45 AM
  3. eigenvalues
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: December 5th 2009, 12:14 PM
  4. Using eigenvectors and eigenvalues to solve DE in matrix form
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: May 8th 2009, 08:09 AM
  5. eigenvalues
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 11th 2009, 12:02 AM

Search Tags


/mathhelpforum @mathhelpforum