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Homework Help: Separation of Variables: How to integrate (x+2y)y'=1 y(0)=2?

  1. Nov 15, 2009 #1
    1. The problem statement, all variables and given/known data
    Use separation of variables to solve (x+2y)y'=1 y(0)=2

    2. Relevant equations
    u=2y+x >>I did not know how to start this, so i looked at the back of the book and it said to use that substitution
    y=(u-x)/2, du=2dy+dx, dy=(du-dx)/2

    3. The attempt at a solution
    so i got the following:


    I could not separate the variables from here. Also, according to the back of the book, the answer is supposed to be 2y-2ln|2+x+2y|+4+2ln2=0. But the term -2ln|2+x+2y| has both x and y variables, so aren't the variables not separated? That still qualifies as a solution by Separation of Variables?
  2. jcsd
  3. Nov 15, 2009 #2


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    Homework Helper

    dx/dx=1, so you last equation is du/dx-1/2=1/u. Can you separate u and x in that? And, no, the equation doesn't separate in y and x, but it does in u and x and I think that counts as a 'separation of variables' solution after the substitution.
  4. Nov 20, 2009 #3
    I finally solved it, thank you!! :smile: :smile: :smile:
  5. Nov 28, 2009 #4
    I got a different answer using linear differential equations.

    dy/dx = 1/(x + 2y)
    dx/dy = x + 2y

    dx/dy - x = 2y

    The answer i got was: x = -2y -2 + 6e^(y-2)

    Differentiating it again returns me to the original differential
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