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Math Help - Differential equation order 4

  1. #1
    Super Member dhiab's Avatar
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    Differential equation order 4

    Solve :

    4y^{\left( 4 \right)} - 23y^{\left( 2 \right)} - y^{\left( 1 \right)} = 0
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by dhiab View Post
    Solve :

    4y^{\left( 4 \right)} - 23y^{\left( 2 \right)} - y^{\left( 1 \right)} = 0
    This is a linear constant coefficient ODE and its charteristic equation has four distinct roots, so what is the problem?

    CB
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  3. #3
    Super Member dhiab's Avatar
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    Quote Originally Posted by CaptainBlack View Post
    This is a linear constant coefficient ODE and its charteristic equation has four distinct roots, so what is the problem?

    CB
    Thank you are you the details?
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  4. #4
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    They're messy. You know one right? Since they're all derivatives, then obviously y=k is a solution and that corresponds to factoring out an m from the characteristic equation: 4m^4-23m^2-m=m(4m^3-23m-1)=0. Here's the others via Mathematica so please excuse the capital letters which Mathematica uses, and sides, I don't have to type the latex this way .

    \left\{m\to \sqrt{\frac{23}{3}} \text{Cos}\left[\frac{1}{3} \text{ArcTan}\left[\frac{2 \sqrt{\frac{3035}{3}}}{3}\right]\right]\right\}

    \left\{m\to -\frac{1}{2} \sqrt{\frac{23}{3}} \text{Cos}\left[\frac{1}{3} \text{ArcTan}\left[\frac{2 \sqrt{\frac{3035}{3}}}{3}\right]\right]+\frac{1}{2} \sqrt{23} \text{Sin}\left[\frac{1}{3} \text{ArcTan}\left[\frac{2 \sqrt{\frac{3035}{3}}}{3}\right]\right]\right\}

    \left\{m\to -\frac{1}{2} \sqrt{\frac{23}{3}} \text{Cos}\left[\frac{1}{3} \text{ArcTan}\left[\frac{2 \sqrt{\frac{3035}{3}}}{3}\right]\right]-\frac{1}{2} \sqrt{23} \text{Sin}\left[\frac{1}{3} \text{ArcTan}\left[\frac{2 \sqrt{\frac{3035}{3}}}{3}\right]\right]\right\}
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by CaptainBlack View Post
    This is a linear constant coefficient ODE and its charteristic equation has four distinct roots, so what is the problem?

    CB
    The characteristic equation is:

    4 \lambda^4 -23 \lambda^2-\lambda=0

    One \lambda factors out to give:

    \lambda ( 4 \lambda^3-23 \lambda -1) =0

    To find the remaining roots (other than \lambda=0) we need the roots of:

    4 \lambda^3-23 \lambda -1 =0

    Which is a depressed cubic and can be solved by Cardarno's trick.

    CB
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