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Math Help - Solving second order ODE

  1. #1
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    Solving second order ODE

    I'm trying to solve a second order ODE in the form of Ay''+By'+1=0. This is part of a genetics problem I have, but I haven't taken DiffEq in a very long time and can't quite remember how to go about solving this. Can anyone help?

    Thanks!
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    Senior Member MaxJasper's Avatar
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    Lightbulb Re: Solving second order ODE

    y(x)= -\frac{a c_1 e^{-\frac{b x}{a}}}{b}-\frac{x}{b}+c_2
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    Re: Solving second order ODE

    Quote Originally Posted by mhguo1 View Post
    I'm trying to solve a second order ODE in the form of Ay''+By'+1=0. This is part of a genetics problem I have, but I haven't taken DiffEq in a very long time and can't quite remember how to go about solving this. Can anyone help?
    Thanks!
    Let f(x)=y'
    This leads to a first order ODE. I suppose that you know the method to solve linear first order ODE.
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  4. #4
    MHF Contributor MarkFL's Avatar
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    Re: Solving second order ODE

    I would write the ODE as follows:

    Ay''+By'=-1

    The characteristic roots are:

    0,-\frac{B}{A} and so the homogeneous solution is:

    y_h(x)=c_1+c_2\frac{A}{B}e^{-\frac{B}{A}x}

    I would next assume a particular solution of the form:

    y_p(x)=kx

    Now, we may use the method of undetermined coefficients to find k:

    y_p'(x)=k

    y_p''(x)=0 and so:

    Bk=-1\,\therefore\,k=-\frac{1}{B} and we have:

    y_p(x)=-\frac{1}{B}x

    By superposition, we now may write the general solution:

    y(x)=y_h(x)+y_p(x)=c_1+c_2\frac{A}{B}e^{-\frac{B}{A}x}-\frac{x}{B}
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  5. #5
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    Re: Solving second order ODE

    Great, thanks for everyone's help!
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