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Finding an implicit, general solution to a homogeonous differential equation

  1. Sep 24, 2011 #1
    1. The problem statement, all variables and given/known data
    Find an implicit, general solution to:

    dy/dx = (6x - 4y) / (x - y) with x > 0.


    2. Relevant equations



    3. The attempt at a solution

    dy/dx = (6x - 4y) / (x - y)

    x(dv/dx) + v = (6 - 4v) / (1 - v)

    [(1-v) / (v-3)(v-2)] dv = dx / x

    [itex]\int dv/(v-2)[/itex] - 2[itex]\int dv/(v-3)[/itex] = [itex]\int dx/x[/itex]

    ln|v-2| -2ln|v-3| + C[itex]_{1}[/itex] = ln|x| + C[itex]_{2}[/itex]

    ln|v-2| -2ln|v-3|- ln|x| = C[itex]_{3}[/itex]

    ----

    Not sure where to go from here, I'm trying to get it into the form

    |y - ax| / (y - bx)^c = C, for some constants a, b, c, and C.

    ----

    Any help is appreciated, thanks.
     
  2. jcsd
  3. Sep 24, 2011 #2

    dynamicsolo

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    Homework Helper

    The integration looks fine. Don't bother bringing everything to one side: you want to leave the function of x on the right, thus ln|v-2| - 2 ln|v-3| = ln|x| + C . Use the properties of logarithms to write the left-hand side as the logarithm of a single expression, then "exponentiate" both sides to eliminate the logarithms. You will have a function x(v) , from which point you could "back-substitute" for v . (I haven't checked yet to see whether this will be nicely-enough-behaved algebraically to get y(x) .)

    EDIT: It isn't, but I see that you were only asked to find an implicit solution involving x and y . That would make sense: the absolute values in the result for v indicate that the solution is a curve which "fails the Vertical Line Test", so it must be split into parts, each of which is the curve for a function.
     
    Last edited: Sep 24, 2011
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