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Math Help - find the general solution

  1. #1
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    find the general solution

    I am suppose to find a small integral root of the characteristic equation by inspection; then factor by division; and find the general solution of

    y^(4) + y^(3) -3y''-5y'-2y=0

    I am having problem with the first part of this trying to find the integral root, can someone please help me??

    Thanks
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  2. #2
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    Quote Originally Posted by mandy123 View Post
    I am suppose to find a small integral root of the characteristic equation by inspection; then factor by division; and find the general solution of

    y^(4) + y^(3) -3y''-5y'-2y=0

    I am having problem with the first part of this trying to find the integral root, can someone please help me??

    Thanks
    So our auxillary equation is

    m^4+m^3-3m^2-5m-2=0

    From the rational roots theorem we know the only possible rational roots are \pm 1 \pm 2

    so we only have 4 things to check

    2^4+2^3-3(2)^2-5(2)-2=0 so m-2 is a factor

    so by long or synthetic division we get

    (m-2)(m^3+3m^2+3m+1)

    we notice that this is the binomial expansion of (m+1)^3

    so we end up with (m-2)(m+1)^3=0

    so the solution is

    y=c_1e^{2x}+c_2e^{-x}+c_3xe^{-x}+c_4x^2e^{-x}
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