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Math Help - When to get rid of absolute value sign in separable differential equations

  1. #1
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    When to get rid of absolute value sign in separable differential equations

    Let's say you have the differential equation:

    dy/dx = ky

    seperation of variables gives:

    dy/y = kdx

    integrating both sides:

    ln |y| = kx + C

    => |y| = Ce^kx

    Now most textbooks seems to just get rid of the absolute value sign, but are you really allowed to do that? I can only imagine it being allowed if you at the same time impose the restriction C > 0 on the free parameter. I'd appreciate any input on this!
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    Re: When to get rid of absolute value sign in separable differential equations

    You're allowed to do that because by definition of an absolute value inequality: If \displaystyle \begin{align*} |x| = a \end{align*}, where \displaystyle \begin{align*} a > 0 \end{align*}, then \displaystyle \begin{align*} x = \pm a \end{align*}.

    So when you have \displaystyle \begin{align*} |y| = C\,e^{k\,x} \end{align*}, that means \displaystyle \begin{align*} y = \pm C \, e^{k\,x} \end{align*}, but a positive or negative constant is still a constant. It's advisable to change the symbol you use for the arbitrary constant after this though...
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    Re: When to get rid of absolute value sign in separable differential equations

    Thanks, I think I get it now! The implicit restriction C>0 is actually there "before" you remove the absolute value, so no need for a restriction when you remove it.
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