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Math Help - 1st order homogeneous ODE

  1. #1
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    1st order homogeneous ODE

    Hello,

    Could you please have a look at this and see if i have done it correctly? thanks!

    (x^2 + y^2) \dfrac{dy}{dx} = 2xy

    \dfrac{dy}{dx} = \dfrac{2xy}{x^2+y^2}


    Would you multiply the numerator and denominator by

    \dfrac{1}{x^2}

    And then given the substitution that

    v = y/x

    dy/dx = x \dfrac{dv}{dx} + v

    So then, putting both equations together,

    \dfrac{2v}{1+v^2} = v + x \dfrac{dv}{dx}

    Rearranging to give:

    \int \dfrac{(1+v^2)}{v(1-v^2)} \, dv = \int \dfrac{1}{x} \, dx

    Then use partial fractions on the first integral:

    \int \dfrac{(1+v^2)}{-v(v+1)(v-1)} \, dv = \int \dfrac{1}{x} \, dx

    \int \dfrac{1}{v} - \dfrac{1}{v+1}- \dfrac{1}{v-1} \, dv = ln|x| + c

    then simply integrate, sub v in, then try to rearrange for y?
    havan't gone homogeneous ODE's before, just trying one out with the method.

    thanks!
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  2. #2
    Member Miss's Avatar
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    Edit: Looks good to me.
    But I did not check your partial fraction.
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  3. #3
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    Quote Originally Posted by Miss View Post
    Edit: Looks good to me.
    But I did not check your partial fraction.
    The partial fraction is also correct.
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