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Wronskian to determine lin.ind. of solutions to a system of ODEs

  1. Mar 23, 2013 #1
    1. The problem statement, all variables and given/known data

    In my book, I'm given that ##\vec{x}_1=\left(\begin{matrix}t^2\\t\end{matrix}\right), \vec{x}_2=\left(\begin{matrix}0\\1+t\end{matrix}\right), \vec{x}_3=\left(\begin{matrix}-t^2\\1\end{matrix}\right)## are solutions. My textbook presents an algebraic way to show that the vectors are linear independent, but I was hoping to see if I can use the Wronskian to show the same result.

    2. Relevant equations

    3. The attempt at a solution

    I thought this was how the Wronskian would look:
    ##W\left(\vec{x}_1, \vec{x}_2, \vec{x}_3\right)=

    \left|\begin{matrix}

    \left(\begin{matrix}t^2\\1\end{matrix}\right) & \left(\begin{matrix}0\\1+t\end{matrix}\right) & \left(\begin{matrix}-t^2\\1\end{matrix}\right)\\

    \left(\begin{matrix}2t\\0\end{matrix}\right) & \left(\begin{matrix}0\\1\end{matrix}\right) & \left(\begin{matrix}-2t\\0\end{matrix}\right)\\

    \left(\begin{matrix}2\\0\end{matrix}\right) & \left(\begin{matrix}0\\0\end{matrix}\right) & \left(\begin{matrix}-2\\0\end{matrix}\right)
    \end{matrix}\right|##

    But I couldn't see how to proceed from there, since matrix multiplication won't work. How would I find this determinant?
     
  2. jcsd
  3. Mar 24, 2013 #2

    vela

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    You can't because it doesn't make sense.

    In any case, you should be able to see by inspection that the three vectors aren't linearly independent.
     
  4. Mar 24, 2013 #3
    Okay, thanks.
     
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