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Math Help - Interval

  1. #1
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    Interval

    Determine the largest interval where there is guaranteed to be a unique solution for the following IVP. Justify why this is the largest interval.

    \textbf{X}'=\left(\begin{array}{cc} \ln t & 6 \\ 3t^2 & \sin t\end{array}\right)\textbf{X} + \left(\begin{array}{c} \sqrt{5-t}\\ \cot t \end{array}\right), \textbf{X}(\frac{5\pi}{2})=\left(\begin{array}{c}0  \\1 \end{array}\right)
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  2. #2
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    Can anyone tell me the best/easiest way to go about finding this interval?
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  3. #3
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    Quote Originally Posted by caeder012 View Post
    Determine the largest interval where there is guaranteed to be a unique solution for the following IVP. Justify why this is the largest interval.

    \textbf{X}'=\left(\begin{array}{cc} \ln t & 6 \\ 3t^2 & \sin t\end{array}\right)\textbf{X} + \left(\begin{array}{c} \sqrt{5-t}\\ \cot t \end{array}\right), \textbf{X}(\frac{5\pi}{2})=\left(\begin{array}{c}0  \\1 \end{array}\right)
    The interval you seek is the largest interval containing the initial point
    on which the derivative is finite and single valued (or assume are allowed to
    make a choice about which branch of \sqrt{} you are on).

    RonL
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  4. #4
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    Quote Originally Posted by CaptainBlack View Post
    The interval you seek is the largest interval containing the initial point
    on which the derivative is finite and single valued (or assume are allowed to
    make a choice about which branch of \sqrt{} you are on).

    RonL
    Alright, well I've put some thought into this problem. I also asked my prof about it. What I found interesting in her reply was that she said I should look at theorems and that I DONT have to solve a system to know whether a unique sol'n exists. Looking at your response, we're given the derivative. That is, X' = ... .

    Looking at theorems in my book, this may be helpful:

    (Note that the form I gave was X' = AX + F(t)).

    So, the theorem says to let the entries of the matrices A(t) and F(t) be functions that are continuous on a common interval I that contains the point t_0. Then there exists a unique sol'n to the IVP X' = A(t)X + F(t) subject to X(t_0) = X_0 on the interval.

    So, I'm not quite sure exactly on how to go about this...

    I decided to look at the functions within the matrix. That is, we know ln(t) is defined for t = (0, infinity). And, 3t^2 is defined for all reals, along with sin(t). Now we look at F(t).

    sqrt(5 - t) is defined for t = (-infinity,5]. And cot(t) is defined (**not really sure.. every multiple of pi/2??)..

    So now we have to see which interval 5pi/2 exists (the largest one?).

    Sorry a little confused. Does my reasoning sound kind of right? I'd appreciate any help.
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  5. #5
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    Well, I'm starting to think no such interval exists. B/c when I tried getting a hint I was told: "Find the interval (if it exists) that has the initial condition AND all the constraints" something along those lines... tricky hard problem. Wish I could just solve this.
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