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Discontinuities of coefficient in linear homogeneous system

  1. Apr 11, 2008 #1
    1. The problem statement, all variables and given/known data

    let [tex]x^{(1)} = \left( \begin{array}{ccc}1\\1\\1\end{array} \right) [/tex], [tex]x^{(2)} = \left( \begin{array}{ccc}1\\t\\t^2\end{array} \right) [/tex], [tex]x^{(3)} = \left( \begin{array}{ccc}1\\t\\t^3\end{array} \right) [/tex]

    a) Find the Wronskian [tex]W(x^{(1)}, x^{(2)}, x^{(3)}) [/tex]

    b) You are told that [tex] x^{(1)}, x^{(2)}, x^{(3)} [/tex] are solutions of a

    linear homogeneous system [tex] x' = P(t)x [/tex]

    i) what can you say about the discontinuities of the coefficient [tex]p_{i,j}(t) [/tex]?

    ii) Find [tex]p_{1,1}(t) + p_{2,2}(t) + p_{3,3}(t) [/tex]. (Hint: use your solution to part a).


    2. Relevant equations
    W = determinant of [tex](x^{(1)}, x^{(2)}, x^{(3)})[/tex]

    3. The attempt at a solution

    For part a), the Wronskian is W = determinant of [tex](x^{(1)}, x^{(2)}, x^{(3)})[/tex], which I found to be [tex] t^4 -2t^3 + t^2 [/tex]. I'm not sure how to do part b). What discontinuities is the question refering to, and how do I find them? How do I use my answer to part a) to solve b)ii)? Any help is appreciated!
     
  2. jcsd
  3. Apr 11, 2008 #2

    HallsofIvy

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    [itex]t^4- 2t^3+ t^2= t^2(t-1)^2= 0[/itex] when t= 0 or t= 1. Since such an equation has a unique solutions as long as the Wronskian is non-zero, what do you think prevents a solution from being extended past t= 0 or t= 1? It must be discontinuities in matrix P.
     
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