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Problem with an ODE

  1. Feb 18, 2005 #1
    [tex]\frac {dy} {dx} = x y^2 - y[/tex]

    I used Mathematica's DSolve function and found the correct answer:
    [tex]y(x) = \frac {1} {1 + x + C e^{x}}[/tex]

    However, I don't have any idea what method to use to solve it with pencil and paper...
  2. jcsd
  3. Feb 18, 2005 #2


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    I cheated,i know;i probably wouldn't have seen it,if you hadn't provided the answer.

    Make the substitution:

    [tex] y(x)=\frac{1}{u(x)} [/tex]

    I believe you'll like the ODE that comes out.

  4. Feb 18, 2005 #3
    I wasn't familiar with the substitution method yet, so I looked it up after reading your post and it looks quite elegant :)

  5. Mar 2, 2005 #4
    Just to put this problem in a general context, it's form is:

    [tex]\frac {dy} {dx} = a(x) y + b(x) y^p[/tex]

    Which is a Bernoulli ODE (or a Ricatti with no constant term).

    The substitution:

    [tex] u(x) = y^{1-p} [/tex]

    reduces this to a first order linear ODE which can be solved in the usual way via an integrating factor.
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