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The Wronskian and the Derivative of the Wronskian

  1. Mar 9, 2012 #1
    1. The problem statement, all variables and given/known data

    y1(t) and y2(t), 2 solutions of the equation:

    y'' +ay'+by=0, with a,b εℝ - {0}
    a) Determine:
    d/dt w(y1,y2)
    where w(y1,y2) is the wronskian of y1(t) and y2(t)
    b)
    Deduce that if (y1(0),y1'(0)^T and (y2(0), y2'(0))^T are 2 linearly independent vectors. Then y1(t) and y2(t) are linearly independent functions.

    2. Relevant equations
    ^T = transpose
    the wronskian is the det |y1 y2|
    |y1' y2'| = y1y2' -y2y1'
    Vectors are linearly independent if w(y1,y2) does not equal 0

    3. The attempt at a solution
    For part a, do I just find the wronskian of y1 and y2 and then take the derivative?
    For part b I'm super confused. I notice that if you transpose the two vectors and put them into a determinant than they are the wronskian.. other than that I'm pretty lost..
     
  2. jcsd
  3. Mar 9, 2012 #2
    for a. yes, although i think you can simplify the result a little...

    for b. start with the given linear combination equal to zero...
     
  4. Mar 10, 2012 #3
    I'm a bit confused regarding the linear combination. Would it just be

    C1V1+C2V2=0?
    I'm not sure what this would accomplish... other than c1v1=-C2V2....
     
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