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Math Help - how do I solve this?

  1. #1
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    how do I solve this?

    dy/dx = (x^2)+(8y^2)/(3xy)? I know that you have to name a parmater v that takes the value y/x and substitute it back in but I'm lost after that.
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  2. #2
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    Quote Originally Posted by lord12 View Post
    dy/dx = (x^2)+(8y^2)/(3xy)? I know that you have to name a parmater v that takes the value y/x and substitute it back in but I'm lost after that.
    What is it you are trying to do to this expression? Integrate it? Differentiate it?
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  3. #3
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    Quote Originally Posted by lord12 View Post
    dy/dx = (x^2)+(8y^2)/(3xy)? I know that you have to name a parmater v that takes the value y/x and substitute it back in but I'm lost after that.
    If your differential equation is

    \frac{dy}{dx} = \frac{x^2 + 8y^2}{3xy}

    You can do a number of things. First the equation is homogeneous so if you let

    v = \frac{y}{x} as you have said, then y = x v then y' = x v' + v, substitute giving

    x v' + v = \frac{x^2 + 8 x^2 v^2}{3x^2 v}

    can x's from the right and solve for v' giving something that is separable. The equation is also Bernoulli.
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  4. #4
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    Ahh. I see.

    Your equation can be written:

     \frac{dy}{dx} = \frac{x^2}{3xy}+\frac{8y^2}{3xy}

    Which gives:


     \frac{dy}{dx} = \frac{x}{3y}+\frac{8y}{3x}

    Giving:

     \frac{dy}{dx} = \frac{1}{3}(\frac{y}{x})^{-1}+\frac{8}{3}\frac{y}{x}

     \text{Let} v = \frac{y}{x}

     y = vx

    Hence

     \frac{dy}{dx} = v +x\frac{dv}{dx}

    Hence:

     v +x\frac{dv}{dx} =  \frac{1}{3}(v)^{-1}+\frac{8}{3}v
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  5. #5
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    how do you write this in terms of x and y?
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  6. #6
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    Quote Originally Posted by lord12 View Post
    how do you write this in terms of x and y?
    First you solve it for v as a function of x. Then you replace v with y/x.
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