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Math Help - Ode .. (2)

  1. #1
    Junior Member BayernMunich's Avatar
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    Ode .. (2)

    Hello
    I need help with the following ODE :

    \dfrac{dx}{(1-xy)^2}+\left[ y^2+\dfrac{x^2}{(1-xy)^2} \right]dy=0

    I multiplied both sides by (1-xy)^2, to get:

    dx+\left[ y^2(1-xy)^2+x^2 \right]dy=0

    Here, I tried the substitution t=xy. But it does not work, doesn't it?

    Any help is appreciated .
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  2. #2
    MHF Contributor
    Jester's Avatar
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    If you write your ODE as

    \dfrac{dx}{dy} + (1-xy)^2y^2+x^2=0
    then this equation is Ricatti. One solution is x = \dfrac{1}{y}.

    If you let x = \dfrac{1}{y} + \dfrac{1}{u}, the equation becomes linear.
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  3. #3
    Junior Member BayernMunich's Avatar
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    Thanks.
    But how did you know this substitution will make it linear?
    By your experience? or there is an method to it?
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  4. #4
    Math Engineering Student
    Krizalid's Avatar
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    Ricatti's equation has de form y'(x)+P(x)y^2+Q(x)y=R(x), then if y_1(x) is a particular solution, then by substituting y(x)=y_1(x)+t you get a Bernoulli equation, but someone just did y(x)=y_1(x)+\dfrac1t and that became the ODE into a linear one.
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