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Math Help - Which Differential Equation Method Do I use?

  1. #1
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    Which Differential Equation Method Do I use?

    i need to solve this differential equation (3y^3-xy)dx-(x^2+6xy^2)dy=0

    I tried using Bernoulli, but it doesn't work. Also tried separating the variables but i cant seem to separate it. i also tried using the 3 cases for exact equation but to no avail. the 3rd case may work but i don't know how to simplify it so i can integrate it.
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  2. #2
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    Quote Originally Posted by lanczlot View Post
    i need to solve this differential equation (3y^3-xy)dx-(x^2+6xy^2)dy=0

    I tried using Bernoulli, but it doesn't work. Also tried separating the variables but i cant seem to separate it. i also tried using the 3 cases for exact equation but to no avail. the 3rd case may work but i don't know how to simplify it so i can integrate it.
    Are you sure it's not (3y^2 - xy) dx -(x^2 + 6xy^2) dy=0?
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    the problem was hand written, so it may have been a mistake. what if it was squared instead of cube? how do you solve it?
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    Quote Originally Posted by lanczlot View Post
    the problem was hand written, so it may have been a mistake. what if it was squared instead of cube? how do you solve it?
    Divide through by y^2. Now read this: Homogeneous Ordinary Differential Equation -- from Wolfram MathWorld

    The technique will be in your classnotes and textbook.
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  5. #5
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    Quote Originally Posted by lanczlot View Post
    i need to solve this differential equation (3y^3-xy)dx-(x^2+6xy^2)dy=0

    I tried using Bernoulli, but it doesn't work. Also tried separating the variables but i cant seem to separate it. i also tried using the 3 cases for exact equation but to no avail. the 3rd case may work but i don't know how to simplify it so i can integrate it.
    If you write the ODE as

    \frac{dy}{dx} = \frac{3y^3-xy}{x^2 + 6xy^2}

    multiply both side by 2y

    2y\frac{dy}{dx} = \frac{6y^4-2xy^2}{x^2 + 6xy^2}

    and let u = y^2 then

    \frac{du}{dx} = \frac{6u^2-2xu}{x^2 + 6xu} (this is homogeneous).
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    wait... how is the differential equation with u homogeneous?
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  7. #7
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    Divide everything on the rhs by x^2 so

    <br />
\frac{du}{dx} = \frac{6\dfrac{u^2}{x^2}-2\dfrac{u}{x}}{1 + 6\dfrac{u}{x}}

    which of the form is \frac{du}{dx} = f\left(\frac{u}{x}\right).
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  8. #8
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    the ODE also admits an integrating factor of the form u(x,y)=x^my^n.
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