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Math Help - help solving differential equation

  1. #1
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    help solving differential equation

    Hi everyone,
    I tried to solve this differential equation, but any method doesnt work me.
    y' = -\frac{x^2y^3 + y}{x^3y^2 - x}.
    I appreciate your help.
    everk.
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  2. #2
    A Plied Mathematician
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    I see some symmetries in your DE of which we might want to take advantage. For example:

    y'=-\dfrac{y}{x}\,\dfrac{x^{2}y^{2}+1}{x^{2}y^{2}-1}.

    I'm thinking a substitution of the form u=yx or u=y/x might be good to try. If we have u=yx, then y=u/x and

    y'=\dfrac{xu'-u}{x^{2}}. It follows, then, that the DE becomes

    \dfrac{xu'-u}{x^{2}}=-\dfrac{u}{x^{2}}\,\dfrac{u^{2}+1}{u^{2}-1}, or xu'-u=-u\,\dfrac{u^{2}+1}{u^{2}-1}. Therefore,

    xu'=u-u\,\dfrac{u^{2}+1}{u^{2}-1}=\dfrac{u^{3}-u-u^{3}-u}{u^{2}-1}=\dfrac{-2u}{u^{2}-1}. The resulting DE xu'=-\dfrac{2u}{u^{2}-1} is separable.

    Conversely, if you try u=y/x, then y=ux and thus y'=u'x+u. The DE becomes

    u'x+u=-u\,\dfrac{x^{4}u^{2}+1}{x^{4}u^{2}-1}. I don't see how this substitution will help you as much as the other one.
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  3. #3
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    Thank you so much for your help. Long time ago I dont resolve any DE.
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  4. #4
    A Plied Mathematician
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    You're very welcome. Have a good one!
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