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ODE with separate variables

  1. Dec 14, 2005 #1
    Hi all,

    here is one ODE I solved now

    [tex]
    y' = 1 + y^2
    [/tex]

    So

    [tex]
    \frac{y'}{1+y^2} = 1
    [/tex]

    [tex]
    \int \frac{dy}{1+y^2} = \int 1 dx
    [/tex]

    [tex]
    \arctan y = x + C \leftrightarrow y = \tan (x + C)
    [/tex]

    [tex]
    x \in (-\frac{\pi}{2} - C, \frac{\pi}{2} - C)
    [/tex]

    The last line is what I'm unsure about.

    Shouldn't it rather be

    [tex]
    x \in (-\frac{\pi}{2} - C + k\pi, \frac{\pi}{2} - C + k\pi), k \in \mathbb{Z}
    [/tex]

    or is it ok as I wrote it originally?

    Thank you.
     
  2. jcsd
  3. Dec 14, 2005 #2

    StatusX

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    Homework Helper

    It depends. Usually you would determine the constant by an initial value, ie, y(x0)=y0, in which case the solution would be valid within whichever region (ie, of width [itex]\pi[/tex]) contains x0. If you're just looking for a completely general form, one that represents every possible solution, then your second form is correct.
     
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