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V substitution in homogeneous equations (diff eq)

  1. Sep 2, 2008 #1
    Hey all, i think i'm doing most of this right, but i'm missing a coefficient somewhere when integrating or something...

    1. The problem statement, all variables and given/known data
    Substitute v=y/x into the following differential equation to show that it is homogeneous, and then solve the differential equation.

    y'=(3y^2-x^2)/(2xy)


    2. Relevant equations
    v=y/x
    y=xv(x) => y'=v+xv'
    v'=dv/dx

    3. The attempt at a solution

    y'=(3y^2-x^2)/(2xy)
    divide top and bottom of right hand side by x^2 to get v's and replace y' by v+xv'
    v+xv'=(3v^2-1)/(2v)
    subtract v from both sides
    xv'=(3v^2-1)/(2v)-v
    put the lonely v on a common denominator
    xv'=(3v^2-1-2v^2)/(2v)=(v^2-1)/(2v)
    separate v's and x's
    (2v)dv/(v^2-1)=dx/x
    integrate
    ln|v^2-1|=ln|x|+c
    substitute v=y/x
    ln|(y/x)^2-1|=ln|x|+c
    simplify
    ln|y^2-x^2|=ln|x|+c


    the back of the book says the answer is |y^2-x^2|=c|x|^3.

    what am i doing wrong? i'm missing a 3 somewhere. i'm kinda rusty with lograthimic algebra, so all help is appreciated. i've gotten a few of these problems wrong by missing a constant or exponent on the right hand of the side of the equation after integrating.
     
  2. jcsd
  3. Sep 2, 2008 #2

    rock.freak667

    User Avatar
    Homework Helper


    These two lines.


    Remember:

    [tex](\frac{y^2}{x^2})-1=\frac{y^2-x^2}{x^2}[/tex]


    Now just simplify again.
     
  4. Sep 2, 2008 #3
    ah, sloppy algebra by me :) thanks.

    then i get
    ln|(y^2-x^2)/x^2|=ln|x|+ln|c|
    ln|(y^2-x^2)|-ln|x^2|=ln|x|+ln|c|
    ln|(y^2-x^2)|=ln|x|+ln|x^2|+ln|c|
    y^2-x^2=c|x|^3

    which is in the back of the book. thanks!
     
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