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Math Help - Solution of system of non-linear equations

  1. #1
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    Solution of system of non-linear equations

    1. Is there a general condition for the existence and uniqueness of solution of a system of simultaneous non-linear equations similar to the determinant test for a system of linear equations.

    2. What are the solution methods (theoretical and numerical) for solving a system of simultaneous non-linear equations.
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  2. #2
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    Re: Solution of system of non-linear equations

    Quote Originally Posted by JulieK View Post
    1. Is there a general condition for the existence and uniqueness of solution of a system of simultaneous non-linear equations similar to the determinant test for a system of linear equations.

    2. What are the solution methods (theoretical and numerical) for solving a system of simultaneous non-linear equations.
    There is no reason to expect a unique solution to a system of simultaneous non-linear equations. Let's take a very simple example.

    $y = ax^2 + c,\ where\ a \ne 0\ and\ a,\ c \in \mathbb R.$

    $y = dx^2 + f,\ where\ d \ne 0,\ d \ne a,\ f \ne c,\ and\ \ d,\ f \in \mathbb R.$

    Find the conditions for real solutions to the system above.

    $0 = y - y = ax^2 + c - (dx^2 + f) = (a - d)x^2 + (c - f) \implies$

    $x = \pm\ \sqrt{\dfrac{f - c}{a - d}}.$

    $No\ real\ solutions\ if\ \dfrac{f - c}{a - d} < 0.$

    $Two\ real\ solutions\ if\ \dfrac{f - c}{a - d} > 0.$

    This system will have either two real solutions or no real solution and will never have a unique real solution.
    Last edited by JeffM; June 15th 2014 at 02:52 PM.
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