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Math Help - differential equations-help!!!!!!!

  1. #1
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    differential equations-help!!!!!!!

    Please help - question attached.
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    Last edited by mr fantastic; February 7th 2009 at 02:42 AM. Reason: Removed the hysterics.
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  2. #2
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    Hello, mbempeni!

    The first one is homegeneous . . .


    \frac{dy}{dx} \;=\;\frac{x^3 + 2x^2y - y^3}{x^3-xy^2}
    Divide top and bottom of the fraction by x^3\!:\quad\frac{dy}{dx} \;=\;\frac{1 - 2\left(\dfrac{y}{x}\right) - \left(\dfrac{y}{x}\right)^3}{1 - \left(\dfrac{y}{x}\right)^2}

    Let v \,=\,\frac{y}{x}\quad\Rightarrow\quad y \,=\,vx\quad\Rightarrow\quad \frac{dy}{dx} \:=\:v + x\frac{dv}{dx}


    Substitute: . v + x\frac{dv}{dx} \;=\;\frac{1-2v-v^3}{1-v^2}


    Then: . x\frac{dv}{dx} \;=\;\frac{1-2v-v^3}{1-v^2} - v \quad\Rightarrow\quad x\frac{dv}{dx}\;=\;\frac{1-3v}{1-v^2}


    Separate variables: . \frac{v^2-1}{3v-1}\,dv \;=\;\frac{dx}{x}


    Can you finish it now?

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  3. #3
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    Quote Originally Posted by Soroban View Post
    Hello, mbempeni!

    The first one is homegeneous . . .

    Divide top and bottom of the fraction by x^3\!:\quad\frac{dy}{dx} \;=\;\frac{1 - 2\left(\dfrac{y}{x}\right) - \left(\dfrac{y}{x}\right)^3}{1 - \left(\dfrac{y}{x}\right)^2}

    Let v \,=\,\frac{y}{x}\quad\Rightarrow\quad y \,=\,vx\quad\Rightarrow\quad \frac{dy}{dx} \:=\:v + x\frac{dv}{dx}


    Substitute: . v + x\frac{dv}{dx} \;=\;\frac{1-2v-v^3}{1-v^2}


    Then: . x\frac{dv}{dx} \;=\;\frac{1-2v-v^3}{1-v^2} - v \quad\Rightarrow\quad x\frac{dv}{dx}\;=\;\frac{1-3v}{1-v^2}


    Separate variables: . \frac{v^2-1}{3v-1}\,dv \;=\;\frac{dx}{x}


    Can you finish it now?
    Can you finish it?It much more difficult than I thought.Please..
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  4. #4
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    Quote Originally Posted by mbempeni View Post
    Can you finish it?It much more difficult than I thought.Please..
    Surely you can integrate the right hand side.

    As for the left hand side, note that \frac{v^2 - 1}{3v - 1} = \frac{v}{3} + \frac{1}{9} - \frac{8}{9} \left( \frac{1}{3v - 1}\right).

    If you're studying differential equations then it's expected that you can apply the various techniques of integration that will be required and which you have no doubt been taught. It might be wise to go back and revise them.
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