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Math Help - Ricatti Equations and combining integration constants.

  1. #1
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    Ricatti Equations and combining integration constants.



    Im all good up until part (d).

    In part (c) we have a DE with constant coefficients with \lambda = 1.

    The general solution for w is
    w(x) = C_1e^x + C_2xe^x
    w'(x) = C_1e^x + C_2e^x + C_2xe^x

    Substituting the formulas for w(x) and w'(x) gives:

    y(x) = e^-x * (C_1e^x + C_2e^x + C_2xe^x) / (C_1e^x + C_2xe^x)

    Simplifying this down gives:

    y(x) = e^-x ( 1 + C_2/(C_1 + C_2x) )

    Now how do I go about combining the two constants to only get one in my final solution? I should only get one since the original DE is only first order.
    Last edited by Corpsecreate; April 23rd 2011 at 08:00 PM.
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  2. #2
    MHF Contributor
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    You lost some stuff in your final answer. I get (unfortunately latex isn't working properly but)

    y = \dfrac{c2(x+1)+c1}{c2x + c1}e^{-x}

    If c2 not = 0 then divide by c2 and call c1/c2 = c, a constant.
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