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Separable ODE, should I take the log of both sides?

  1. Jan 13, 2010 #1
    1. The problem statement, all variables and given/known data
    Solve dy/dx = exp(x-y) given that y = ln 2 at x = 0


    2. Relevant equations
    None.


    3. The attempt at a solution

    Firstly let's get the equation in to a form so we can re-arrange the x's and y's, and then re-arrange.

    dy/dx = exp(x)/exp(y)

    exp(y)*dy = exp(x)*dx

    Integrate:

    exp(y) = exp(x) + C

    Substitute:

    exp(ln(2)) = exp(0) + C

    2 = 1 + C, therefore C = 1

    Hence, exp(y) = exp(x) + 1, and take log to get an equation in the form of y = ...

    ln(exp(y)) = ln(exp(x)) + ln(1) =>

    y = x

    Doesn't really look right though, am I right or wrong?

    Edit: Solved-> ln(exp(y)) = ln(exp(x) + 1)
    => y(x) = ln(exp(x) + 1)
     
    Last edited: Jan 13, 2010
  2. jcsd
  3. Jan 13, 2010 #2

    vela

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    Looked OK until the last step when you took the logs. log(a+b) isn't equal to log(a)+log(b).

    Note also that the initial condition no longer holds if y=x was the solution.
     
  4. Jan 13, 2010 #3
    I solved the problem using Maple; it gives the answer as y = ln(exp x + 1).

    So I guess I'm meant to take the log of the entire expression on the r.h.s.
     
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