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Help Solving a Nonlinear ODE

  1. Feb 6, 2010 #1
    Hello everyone and thanks for looking at my thread,

    I had some trouble solving this ODE which was in a textbook by Henry J. Ricardo:

    [tex]x(e^y - y') = 2[/tex].

    This problem is from a section dealing with linear equations, but there is a hint beside the problem which reads, "Hint: Think of y as the independent variable, x as the dependent variable, and rewrite the equation in terms of dx/dy."

    I tried to solve it by doing the following:

    [tex]e^y - y' = 2 / x[/tex]
    [tex]y' = e^y - 2 / x[/tex]
    [tex]y' = (xe^y - 2) / x[/tex]
    [tex]dx/dy = x / (xe^y - 2)[/tex]
    [tex]xe^y + e^y (dx/dy) = x[/tex]
    [tex]dx/dy + x = xe^{-y}[/tex]
    [tex]dx/dy = x(e^{-y} - 1)[/tex]
    [tex](1 / x)dx = (e^{-y} - 1)dy[/tex]
    [tex]ln|x| + C = -e^{-y} - y[/tex]

    However, I know that this is not the correct solution; I'm guessing that I went wrong somewhere when I rewrote the equation in terms of [tex]dx/dy[/tex], but I don't know how else I would approach the problem. Where did I go wrong with my work, and how would I proceed to solve the problem correctly?

    Thank you very much for your help!
    Last edited: Feb 6, 2010
  2. jcsd
  3. Feb 7, 2010 #2
    I think you made an error when going from line 4 to line 5?. Line 4 starting with dx/dy is ok. But the next line doesn't seem right? I haven't check in detail, only quickly in my head, so who knows...

  4. Feb 10, 2010 #3
    I didn't look at your solution, but if you want to solve it just substitute e^y = z and it becomes the Bernoulli equation.
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