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Thread: Simple but conceptual question

  1. #1
    Junior Member
    Nov 2007

    Simple but conceptual question

    So I may just be rusty on my algebra, but I'm covering differential equations at a basic level right now and I've seen this pop up a few times and I'd love to know the proof behind why.

    I'll see either $\displaystyle 10-x=e^{-kt-c1}$ become $\displaystyle x=10-Ce^{-kt}$ or say $\displaystyle y+4=e^{\frac{-x^2}{2}+C1}$ become $\displaystyle y+4=Ce^{\frac{-x^2}{2}}$

    I just don't quite see how they're pulling that C down and in front of the exponential function...can anyone offer a proof or other explanation as to why? Thanks!
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  2. #2
    Super Member Aryth's Avatar
    Feb 2007
    In the equations you're giving, they are using little c's... The big C stands for:

    $\displaystyle C = \pm e^{c_1}$

    Or e to the power of some constant... Since that does not change it's constancy, it's valid.

    In most basic differential equations, the C becomes the initial value anyway.

    For example:

    $\displaystyle 10 - x = e^{-kt - c_1}$

    $\displaystyle 10 - x = e^{-kt}e^{-c_1}$

    $\displaystyle C = e^{-c_1}$

    $\displaystyle 10 - x = Ce^{-kt}$

    $\displaystyle 10 - Ce^{-kt} = x$
    Last edited by Aryth; Apr 18th 2008 at 10:17 PM.
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