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Math Help - Third order DE

  1. #1
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    Third order DE

    2y^(^3^) + 9y" + 12y' + 5y =0
    Tried using an auxiliary equation:
    2m^3 + 9m^2 + 12m + 5 = 0

    Not sure how to factor that so I don't know where to go from there, I feel like I might be missing something obvious, any help is appreciated
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  2. #2
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    Quote Originally Posted by Naples View Post
    2y^(^3^) + 9y" + 12y' + 5y =0
    Tried using an auxiliary equation:
    2m^3 + 9m^2 + 12m + 5 = 0

    Not sure how to factor that so I don't know where to go from there, I feel like I might be missing something obvious, any help is appreciated
    You need to use the rational root theorem.

    Since all the terms are positive the roots must be negative and by the rational root theorem the only possible roots are

    -1,-5,-\frac{1}{2},-\frac{5}{2}

    If you check negative one you will find that it is a root so x+1 is a factor by long division you get

    (m+1)(2m^2+7m+5)
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