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Absolute value in separable ODEs?

  1. Dec 20, 2013 #1
    Suppose I have a variable separable ODE, e.g.,
    [tex] \frac{dy}{dx} = 3y. [/tex]
    We all know that the solution is [itex]y=Ae^{3x}[/itex] where A is a constant. My question is as follows. To actually find this solution we rearrange the equation and integrate to get
    [tex] \int \frac{dy}{y} = 3 \int dx, [/tex]
    which gives
    [tex] \ln |y| = 3x + C [/tex] where C is a constant. I would have thought that this gives the solution
    [tex] |y| = Ae^{3x} \qquad \mbox{where} \qquad A=e^C. [/tex]
    My question is how can we get rid of the absolute value sign in the actual answer? Is it because A = e^C must always be positive? But how come in general that does not have to be true for the ODE to be satisfied?
     
  2. jcsd
  3. Dec 20, 2013 #2

    HallsofIvy

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    It is precisely for that reason that you must have the absolute value on y.

    Yes, [itex]|y|= Ae^x[/itex] where [itex]A= e^C[/itex] is positive.

    Therefore [itex]y= Ae^x[/itex] or [itex]y= -Ae^{x}[/itex].
     
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