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Math Help - diff equation problem help needed

  1. #1
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    diff equation problem help needed

    how do i solve this problem here? someone please help me

    (2y^2+5x)dx+3xydy=0 solve
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  2. #2
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    Quote Originally Posted by uga75 View Post
    how do i solve this problem here? someone please help me

    (2y^2+5x)dx+3xydy=0 solve
    This equation can be made exact by finding an integrating factor

    Let \phi(x) be a differentable function of x

    multiplying the equation by phi we get

    (2y^2+5x)\phi(x) dx+3xy \phi(x) dy=0

    Remember to be exact that if we have an eqaution of the form

    Mdx+Ndy=0 it is exact if

    \frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}

    4y\phi(x)=3x\phi(x)+3xy\frac{d\phi}{dx} \iff y\phi(x)=3xy\frac{d\phi}{dx} \iff \frac{1}{x}\cdot \frac{1}{x}dx=\frac{1}{\phi}d\phi

    Now solving for phi we get


    \frac{1}{3}\ln(x)=\ln(\phi) \iff \phi(x)=x^{1/3}

    So using this we get

    (2x^{1/3}y^2+5x^{4/3})dx+3x^{4/3}ydy=0

    This is now exact and can be solve using that method.

    I hope this helps.
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  3. #3
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    Quote Originally Posted by uga75 View Post
    how do i solve this problem here? someone please help me

    (2y^2+5x)dx+3xydy=0 solve
    You will notice that written in standard form

    \frac{dy}{dx} + \frac{2}{3} \frac{y}{x} = - \frac{5}{3y}

    the ODE is Bernoulli.
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