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Homework Help: An eigen-What?

  1. Apr 15, 2006 #1
    Given:Second order ODE: x" + 2x' + 3x = 0
    Find:
    a) Write equation as first order ODE
    b) Apply eigenvalue method to find general soln

    Solution:

    Part a, is easy
    a) y' = -2y - 3x

    now, how do I do part b? Do I solve it as a [1x2] matrix?
     
  2. jcsd
  3. Apr 16, 2006 #2
    I don't think you have part a quite correct. It believe should be a matrix equation, something like z' = Az, where z is vector and A is a 2x2 matrix. You would then use the eigenvalue method on the 2x2 matrix.
     
  4. Apr 16, 2006 #3
    Your solution of part a is wrong. I think you should define the vector
    [tex] u=\left(\begin{array}{cc}x'\\x\end{array}\right) [/tex]
    so the derivative of u:
    [tex] u'=\left(\begin{array}{cc}x''\\x'\end{array}\right) [/tex]
    By substituting x''=- 2x' - 3x into u'=(x'';x'), you get:
    [tex] u'=\left(\begin{array}{cc}- 2x' - 3x\\x'\end{array}\right) [/tex]
    and will easily find the solution, something like u' = Au + B as eigenglue said.
     
    Last edited: Apr 16, 2006
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