1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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:

    dy/dx=1/(x+2y)
    (du-dx)/(2dx)=1/(x+2(u+x)/2)
    (du-dx)/(2dx)=1/u

    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

    Dick

    User Avatar
    Science Advisor
    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
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook