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Math Help - Integration using separation of variables

  1. #1
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    Integration using separation of variables

    Integrate the following integral using separation of variables :

    vx = (1/3)v -(8/9) (vx is the partial derivative of v w.r.t x) and v is a function of two variables.

    Can someone please show me the steps involved for separating the variables for this question ?

    Please Help.
    Last edited by mrmaaza123; March 31st 2013 at 02:37 AM.
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  2. #2
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    Re: Integration using separation of variables

    How many variables is v a function of?
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    Re: Integration using separation of variables

    v is a function of two variables , x and y
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    Re: Integration using separation of variables

    Since y does not appear in the equation, treat as an equation in x only.
    \frac{dv}{dx}= \frac{1}{3}(v- \frac{8}{3})
    \frac{dv}{v- \frac{8}{3}}= \frac{dx}{3}
    Integrating
    ln\left|v-\frac{8}{3}\right|= \frac{x}{3}+ f(y)
    where the "constant of integration" can be any function of y only.

    We can solve that for v(x, y)= \frac{8}{3}+ e^{x/3}F(y)
    where, since f(y) can be an arbitrary function of y, F(y)= e^{f(y)} is also an arbitrary function of y.
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  5. #5
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    Re: Integration using separation of variables

    But in my book the answer is v(x,y) = 8/3 +(1/3)e^x/3 F(y).
    Where do you think that extra factor of 1/3 came from ? or is it incorrect ?
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