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Math Help - existence and uniqueness of nth-order IVP solutions

  1. #1
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    existence and uniqueness of nth-order IVP solutions

    Hi guys. I'm given the following:

    Determine the existence and uniqueness of the solutions of the initial value problem (IVP)

    y'''=(\sin t)e^y+y^2+(y')^{4/3},

    y(t_0)=a_0, y'(t_0)=a_1, y''(t_0)=a_2.

    I really don't know where to begin on this one. Any help would be much appreciated!
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  2. #2
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    Let y0=y, y1= y', y2= y''. They y'''= y2' so the equation says y1'= sin(t)e^{y0}+ (y0)^2+ (y1)^{4/3} and y0'= y1. You can write those two equations as the single matrix equation
    \begin{bmatrix}y0 \\ y1\\ y2 \end{bmatrix}'= \begin{bmatrix}y1 \\ y2 \\ sin(t)e^{y_0}+ (y_0)^2+ (y_1)^{4/3}\end{bmatrix}
    with initial condition \begin{bmatrix}y0(t_0) \\ y1(t_0) \\ y2(t_0)\end{bmatrix}=\begin{bmatrix}a_0 \\ a_1 \\ a_2\end{bmatrix}

    Now do you know the conditions for existence and uniqueness of solutions to a first order equation? Apply them to this equation.
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  3. #3
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    Thanks for the starting point, but I'm still pretty lost. I've never done an exercise like this before, so it's giving me some real trouble!

    We have the IVP y'=f(y,t), where y=(y_0,y_1,y_2)^T. If it were just a basic real-valued function (instead of a vector-valued function), then I'd check for local Lipschitz continuity, maybe by taking the partial derivative f_y(y,t), and seeing whether or not it is bounded, or else straight from the definition. But I have no idea how to look for Lipschitz continuity in a vector-valued function.

    Thanks again for the help so far!
    Last edited by hatsoff; September 8th 2010 at 04:15 PM.
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