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First Order Diff Eq

  1. Jan 17, 2009 #1
    1. The problem statement, all variables and given/known data

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


    2. Relevant equations



    3. The attempt at a solution

    Let u = 2x + y
    du = 2 + dy/dx
    dy/dx = du/dx - 2

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

    1/sqrt(2) * arctan ((u-1)/sqrt(2)) = x + c

    1/sqrt(2) * arctan ((2x + y -1)/sqrt(2)) = x + c

    y = sqrt(2) * tan(sqrt(2) * x + C) + 1 - 2x

    BUT the answer in the back of the textbook is y^2 = x^2 + cx^3. Did I mess up or is it a typo? Thank you.
     
  2. jcsd
  3. Jan 17, 2009 #2

    Mark44

    Staff: Mentor

    You have two answers. Check them to see if they satisfy dy/dx = (2x + y - 1)^2. If you find that your answer satisfies this DE, that's pretty good evidence that the book's answer is wrong. I don't see anything obviously wrong with your work.
     
  4. Jan 17, 2009 #3
    Cool thanks my solution worked.
     
  5. Jan 17, 2009 #4

    Mark44

    Staff: Mentor

    If you feel really ambitious, you could check the book's solution. Sometimes with differential equations it's possible to get what look like completely different solutions, but they both work. The key is that they differ by a constant.

    As an example, sin^2(x) and -cos^2(x) look to be very different, but differ only by a constant.
     
  6. Jan 17, 2009 #5
    Yep the book's sol'n sure doesn't work. Thanks for the help! :)
     
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